Elementary Particles Physics Textbook: Griffiths, 2nd Ed.

PHYSICS TEXTBOOK
David Griffiths
,vWILEY-YCH
Introduction to
Elementary Particles
Second, Revised Edition
David Griffiths
Introduction to Elementary Particles
Second, Revised Edition
~
WILEY­
VCH
WllEY-VCH Verlag GmbH & Co. KGaA
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I'
Contents
Preface to the First Edition IX
Preface to the Second Edition XI
Formulas and Constants XIII
Introduction
1.10
1.11
Historical Introduction to the Elementary Particles 13
The Classical ERA (1897-1932) 13
The Photon (1900-1924) 15
Mesons (1934-1947) 18
Antiparticles (1930-1956) 20
Neutrinos (1930-1962) 23
Strange Particles (1947-1960) 30
The Eightfold Way (1%1-1964) 35
The Quark Model (1964) 37
The November Revolution and Its Aftermath (1974-1983 and
1995) 44
Intermediate Vector Bosons (1983) 47
The Standard Model (1978-?) 49
2
2.1
2.2
2.3
2.4
2.4.1
2.4.2
2.4.2.1
2.4.3
2.4.4
2.5
2.6
Elementary Particle Dynamics 59
The Four forces 59
Quantum Electrodynamics (QED) 60
Quantum Chromodynamks (QeD) 66
Weak Interactions 7J
Neutral 72
Charged 74
Leptons 74
Quarks 75
Weak and Electromagnetic Couplings of Wand Z
Decays and Conservation Laws 79
Unification Schemes 84
1.1
12
1.3
lA
1.5
1.6
17
1.8
1.9
78
Inlmd"<lio� 10 &mtnla'1' PArfirks. Stumd Edilio�. D.vid Griffiths
Copyright 0 2008 WILEY-VCH Verlag GmbH & Co. KGaA. Weinheim
1SON, 978-)·527-4060H
VII COlllenll
1
Relativistic Kinematics 89
Lorentz Transformations 89
Four-vectors 92
Energy and Momentum 96
Collisions 100
Classical Collisions 100
Relativistic Collisions 101
Examples and Applications 102
•
Symmetries 115
Symmetries. Groups. and Conservation Laws
Angular Momentum 120
Addition of Angular Momenta 122
Spin t 125
Flavor Symmetries 129
Discrete Symmetries 136
Parity 136
Charge Conjugation 142
CP 144
Neutral Kaons 145
CP Violation 147
Time Reversal and the TCP Theorem 149
S
Bound States 159
The Schrodinger Equation 159
Hydrogen 162
Fine Structure 165
The Lamb Shift 166
Hyperfine Splitting 167
Positronium 169
Quarkonium 171
Charmonium 174
Bottomonium 175
Light Quark Mesons 176
Baryons 180
Baryon Wave Functions 181
Magnetic Moments 189
Masses 191
3.1
3.2
3.3
3.4
3.4.1
3.4.2
3.5
'.1
4.2
4.2.1
4.2.2
'.3
4.4
4.4.1
4.4.2
4.4.3
4.4.3.1
4.4.3.2
4.4.4
5.1
5.2
5.2.1
5.2.2
5.2.3
5.3
5.4
5.4.1
5.4.2
5.5
5.6
5.6.1
5.6.2
5.6.3
6
6.1
6.1.1
6.1.2
6.2
6.2.1
The Feynman Calculus 197
Decays and Scattering 197
Decay Rates 197
Cross Sections 199
Tbe Golden Rule 203
Golden Rule for Decays 204
115
Contents
6.2.1.1
6.2.2
6.2.2.1
6.3
6.3.1
6.3.2
6.3.3
Two-particle Decays 206
Golden Rule for Scattering 208
Two-body Scattering in the CM Frame 209
FeynmanRules for aToyTheory 211
Lifetime of theA 214
A + A -+ B + B Scattering 215
Higher-order Diagrams 217
7
Quantum Electrodynamics
7.1
7.2
7.3
7.4
7.S
7.6
7.7
7.8
7.9
The Dirac Equation 225
Solutions to the Dirac Equation 229
Bilinear Covariants 235
The Photon 238
The Feynman Rules for QED 241
Examples 245
Casimir's Trick 249
Cross Sections and Lifetimes 254
Renormalization 262
8
Electrodynamics and Chromodynamics ofQuarks
8.1
8.2
8.3
8.4
8.4.1
8.4.2
8.S
8.6
Hadron Production in e+e- Collisions 275
Elastic Electron-Proton Scattering 279
FeynmanRules For Chromodynamics 283
Color Factors 289
Quark and Antiquark 289
Quark and Quark 292
Pair Annihilation in QCD 294
Asymptotic Freedom 298
9
9.1
9.2
9.3
9.4
9.S
9.6
9.7
9.7.1
9.7.2
9.7.3
Weak Interactions 307
Charged leptonic Weak Interactions 307
Decay of the Muon 310
Decay of the Neutron 315
Decay of the Pion 321
Charged Weak Interactions of Quarks 324
Neutral Weak Interactions 329
Electroweak Unification 338
Chiral Fermion States 338
Weak Isospin and Hypercharge 342
Electroweak Mixing 345
10
Gauge Theories
10.1
10.2
10.3
225
275
353
Lagrangian Formulation of Classical Particle Mechanics
Lagrangians inRelativistic FieldTheory 354
Local Gauge Invariance 358
353
I
VII
VillI COlllell!!
IDA Yang-Mills Theory 361
10.5
10.6
10.7
10.8
10.9
Chromodynamics 366
Feynman Rules 369
The Mass Term 372
Spontaneous Symmetry·breaking
The Higgs Mechanism 378
,1
Neutrino Oscillations 387
The Solar Neutrino Problem
Oscillations 390
Confirmation 392
Neutrino Masses 395
The Mixing Matrix 397
11.1
11.2
11.3
11A
11.5
12
375
387
12.1
12.2
12.3
12A
12.4.1
12.4.2
12.5
12.5.1
12.5.2
12.6
Afterword: What's Next? 401
The Higgs Boson 401
Grand Unificati on 405
Matter/Antimatter Asymmetry 409
Supersymmetry, Strings, Extra Dimensions
Supersymmetry 411
Strings 413
Dark Matter/Dark Energy 414
Dark Matter 414
Dark Energy 416
Conclusion 417
A
The Dirac Delta Function
B.l
B
B.1.1
B.2
B.2.1
C
C.l
C.2
D
0.1
0.2
0.3
423
Decay Rates and Cross Sections
Decays 429
Two-body Decays 429
Cross Sections 430
Two-body Scattering 430
Pauli and Dirac Matrices
Pauli Matrices 433
Dirac Matrices 434
433
Feynman Rules (Tree Level)
External Lines 437
Propagators 437
Vertex Factors 438
Index 441
437
429
411
I"
Preface to the First Edition
This introduction to the theory of elementary particles is intended primarily for
advanced undergraduates who are majoring in physics. Most of my colleagues
consider this subject inappropriate for such an audience - mathematically too
sophisticated. phenomenologically too cluttered, insecure in its foundations. and
uncertain in its future. Ten years ago I would have agreed. But in the last decade the
dust has settled to an astonishing degree. and it is fair to say that elementary particle
physics has come of age. Although we obviously have much more to learn, there
now exists a coherent and unified theoretical structure that is simply too exciting
and important to save for graduate school or to serve up in diluted qualitative form
as a subunit of modern physics. I believe the time has come to integrate elementary
particle physics into the standard undergraduate curriculum.
Unfortunately, the research literature in this field is clearly inaccessible to
undergraduates, and although there are now several excellent graduate texts, these
call for a strong preparation in advanced quantum me<:hanics, if not quantum
field theory. At the other extreme, there are many fine popular books and a
number of outstanding
SCientifo; Amtrican articles. But very little has been written
specifically for the undergraduate. This book is an efo
f rt
out of a one-semester elementary particles course I have taught from time to
time at Reed College. The students typically had under their belts a semester of
electromagnetism (at the level of lorrain and Corson), a semester of quantum
mechanics (at the level of Park), and a fairly strong background in spe<:ial relativity.
In addition to its principal audience, I hope this book will be of use to beginning
graduate students, either as a primary text, or as preparation for a more sophisticated
treatment. With this in mind, and in the interest of greater completeness and
flexibility, I have included more material here than one can comfortably cover in a
single semester. (In my own courses I ask the students to read Chapters 1 and 2
on their own, and begin the le<:tures with Chapter 3. I skip Chapter 5 altogether,
concentrate on Chapters 6 and 7, discuss the first two Sl'(tions of Chapter S, and
then jump to Chapter 10.) To assist the reader (and the teacher) I begin each
chapter with a brief indication of its purpose and content, its prerequisites, and its
role in what follows.
)"m./ucIiOll '" &....."''1' P�..w... S«<md Edil;"'" D.vid Griffith.
CopyrightCl2008 WILEY·VCH Ve-r48 GmbH & Co. KGaA. W";nh,,;m
ISBN: 97!-J·Sli·.06OH
x
I Prtfou 10 Ihe Firsl Edmon
This book was written while I was on sabbatical at the Stanford linear Accelerator
Center, and I would like to thank Professor Sidney Drell and the other members of
theTheory Group for their hospitality.
DAVID GRIFFITHS
1986
I"
Preface to the Second Edition
It is 20 years since the first edition of this book was published, and it is both
gratifying and distressing to reflect that it remains, for the most part, reasonably
up-to-date. There are, to be sure, some gross lacunae - the existence of the top
quark had not been confirmed back then, and neutrinos were generally assumed
(for no very good reason) to be massless. But the Standard Model, which is, in
essence, the subject of the book, has proved to be astonishingly robust. This is
a tribute to the theory, and at the same time an indictment of our collective
imagination. ! don't think there has been a comparable period in the history of
elementary particle physics in which so little of a truly revolutionary nature has
occurred. What about neutrino oscillations? Indeed: a fantastic story (I have added a
chapter on the subje<I); and yet, this extraordinary phenomenon fits so comfortably
into the Standard Model that one might almost say, in retrospect (of course), that it
would have been more surprising if it had not been so. How about supersymmetry
and string theory? Yes, but these must for the moment be regarded as speculations
(I have added a chapter on contemporary theoretical developments). As far as solid
experimental confirmation goes, the Standard Model (with neutrino masses and
mixing) still rules.
In addition to the two new chapters already mentioned, I have brought the
history up·to-date in Chapter 1. shortened Chapter S, provided (I hope) a more
compelling introduction to the Golden Rule in Chapter 6, and eliminated most of
the old Chapter 8 on electromagnetic form factors and scaling (this was crucially
important in interpreting the deep inelastic scattering experiments that put the
quark model on a secure footing, but no one today doubts the existence of quarks,
and the technical details are no longer so essential). What remains of Chapter 8
is now combined with the old Chapter 9 to make a new chapter on hadrons_
Finally, have prepared a complete solution manual (available free from the
publisher, though only - 1 regret - to course instructors). Beyond this the changes
are relatively minor.
Many people have sent me suggestions and corrections, or patiently answered m y
questions. 1 cannot thank everyone, but I would like to acknowledge some o f those
who were especially helpful: Guy Blaylock (UMass), John Boersma (Rochester),
Carola Chinellato (Brazil), Eugene Commins (Berkeley), Mimi Gerstell (Cal Tech),
I
/ntr(ldUCIi"" I<J Elemenl<Jry Parlicl<$, S«<>nd &/ili()IL o...;d Griffiths
CopyrightC> 2008 WILEY.vCH V�ria8 GmbH & Co. KGaA, W�inMim
ISBN: 978·J.52H0601·2
xu
I Prtfau 10 Ihe �cOMd EdiljOM
Nahmin HOIwitz (Syracuse). Richard Kass (Ohio State). Janis McKenna (UBC).
Jim Napolitano (RPI). Nic Nigro (Seattle). John Norbury (UW.Milwaukee). Jason
Quinn (Notre Dame). Aaron Roodman (SLAC). Natthi Shanna (Eastern Michigan).
Steve Wasserbeach (Haverford). and above all Pat Burchat (Stanford).
Part of this work was carried out while I was on sabbatical. at Stanford and
SLAe. and I especially thank Patricia Burchat and Michael Peskin for making this
possible.
DAVID GRIFFITHS
ZOO,
I
Formulas and Constants
XIII
Particle Data
Mass in MeV/c?lifetime in seconds. charge in units of the proton charge.
Leptons (spin 1/2)
Generation
fin!
second
third
Fbw,
t (electron)
Charge
Mass'
I
0.510999
v, (e neutrino)
0
0
jl (muon)
�� (p. neutrino)
I
105.659
(tlU)
�, (T neutrino)
I
0
r
0
Ufetime
00
00
In6.99
,��ii,
2.91 x 10-
'v,v,.jl�,ii�.1f "
00
0
-
2.19703 x 1000
0
Principal Deays
-
-
-
'Neutrino masses are extremely small. and for most purposes an be taken to be zeTO; for details see Chapter 11.
Quarks (spin 1/2)
Flavor
ChaT�
Mass'
fi�,
d (down)
III
7
second
s (strange)
Generation
u (up)
b (bottom)
, (warm)
third
I (top)
'I'
,
II'
120
'I'
1200
II'
4]00
'I'
174000
'Ught quark masses are imprecise and specul.tive; for effective
Mediators (spin 1)
Force
Mediator
Charge
Mass'
Strong
g (8 gluons)
0
Electromagnetic
r (photon)
0
Weak
zfJ (neutral)
W± (charged)
0
0
±I
80,420
0
91.l90
Lifetime
00
00
1.11 x lO-l)
2.64 x 10-15
Princip.ol Decays
-
,''v•• jl..�f'. T+V,. eX .... hadrons
,te-. jltjl-. rtr-. qq .... hadrons
Baryons (spin 1/2)
N
Quark Content
Charge
Mass
Ufetime
Princip;il DeclYs
1
0
938.272
939.565
885.7
peu,
E'
E-
""'
0
1
0
dd,
-I
S-
k
-I
1115.68
1189.37
1192.64
1197.45
1314.8
1321.3
2286.S
Baryon
I�
"""
"'"
A
"'"
E'
"�
a'
""
A'
,
0
1
""
""
2.G3 x 10-10
8.02 x IO-ll
7.4 x 10-20
lA8 x 10-10
2.90 x 10-10
1M x 10- 10
2.00 x 1O- 11
Baryons (spin 3/2)
Baryon
Qll;;rk Content
Charge
Mass
•
uuu, uud, auld, dad
S'
UllS, lids, ads
uss, ass
2.1,0, 1
1,0,-1
0,-1
1232
1385
1533
m
-I
1672
E'
0-
Quark Content
Meson
.'
Charge
Mass
I, 1
139.570
ui, jU
0
1,-1
134.9n
493.68
ds, sd
0
497.65
(uli+ ad +5S);JJ
0
547.51
0
957.78
cd,&
1,-1
1869.3
cu, uC
0
1,-1
1864.5
1%8.2
1.-1
sm.o
(UU - a4-,;,(i
K'
,
K',K'
I ,
! ,
I I>'
,
; cP, ""&
I 0;
juli + ad - 2sr,j./6
lib, "Ii
db, bd
cS,�
I If,"I
i
B'
K'
p
0
lUI, (1111 MJ/J_2, au
Quark Content
uS, as. sci, sii
W
(uu + ddJ;,;1
D'
ed, CU, lIZ, &
"
T
"
bi
A.'
A.pKIf, Alflf, Elflf
Prind� DecilYS
Ufetime
5.6 x to1.8 x 10-2 )
6.9 x 10-2J
8.2 x 1O-11
N.
AIf,I:1f
3lf
AK-, 311"
lifetime
{�:
1.24 x 1O-�
3.2 x 10-/ 1
1.04 x 10- 11
4.1 x 10-])
5.0 x l O-u
1.6 x 10-11
I.S x 10-12
Chilfge
Mass
1,0,.1
775.5
1,0,·1
89'
0
0
1,0..1
782.G
3097
2008
9460
"',
yy
8.95 x 10-11
K1: 5.11 x 1O-�
5.1 x 10-19
5279.4
0
Princip;il Decays
2.60 x 104
8.4 x 10-11
Vector Mesons (spin 1)
Meson
Ay
M-
Pseudoscalar Mesons (spin 0)
JUI, au
.'
P1l"-, 1"\11"9
P1f9, 1"\11"+
lifetime
4 x 1O-2�
1 x 10-11
8 x 10-23
7 x 10-21
3 x 10-/1
1 X 10-20
jtv,., Iflf, Iflflf
••
Ifeu" IfjtV,., 1I"lflf
yy, Iflflf
qlflf, PY
Klflf, Kjtv,., }(ev,
IJP, �lflf, ¢p
D'tv/, Dtv" D'lflflf
Klflf, }(ev" Kjtv,.
D'lVt, Dlvt, D'lflf
Prind� Decays
""
K.
1I"lflf,lfy
e+e-, jt+jt-, 5lf, 7lf
Dlf, Dr
ete-, 1(+1(-, r+r-
I'"
Spin 1/2
Pauli Matrices:
aiOj = �y + �ij�O'\,.
ol =0;=(1;-1,
Dirac Matrices:
Y ..
"
(
)
1
0
0
-1 '
.
cr' !!
(<I' O')(b· u) "" a·b + itT . (a x b)
ei8·Q = cos{J+i(�·O")sin{J
( 0o;) '
0
-(1j
o
o
-1
o
(For product rules and trace theorems see Appendix C.)
Dirac Equation:
fJI-mc)u=O,
f/+ mc)v =0,
;j"",,,,tyO,
ull-mc) =0.
r""yOrtyO, I-a"y"
Feynman Rules
Propagators
External Lines
SpinO:
Nothi ng
Incoming particle:
Spin 1/2:
v(f+mc)=O
Incoming antiparticle:
Outgoing particle:
Outgoing antiparticle:
"
"
"
"
ill+me)
" (mc)2
"'" I
Spin 1:
I Incoming: �
1 Outgoing; �;
..
-i{&,v - q",qv/(mc)z]
q2
(me)2
(For vertexfactors see Appendix D.)
Fundamental Constants
Planck's constant:
=
•
1.05457 X 10-34 Js
6.58212 X 10-22 MeVs
,
Speed oflighl:
Mass of electron:
rn,
Mass of proton:
m,
Electron charge (magnitude):
,
2.99792 x 108mls
=
9.10938 x IO-llkg = 0.510999 MeV/c2
1.67262 x IO-27kg = 938.272 MeV/c!
1.60218 x 10-19 C
4.80320 X 10-10esu
Fine structure constant:
a
Bohr radius:
"
Bohr energies:
E.
Classical electron radius:
'.
QED coupling constant:
�
Weak coupling constants:
g.
g,
Weak mixing angle:
O.
Strong coupling constant:
�
illhc = 1/137.036
hZ/mer = 5.29177 x lO-ll m
=
=
=
=
=
=
0.1nm = 10-10m
Ifm
10-15 m
I barn
10-28 m2
I,V
1.60218 X 10-19 ,
1 Coulomb
c-/m,;Cl = 2.81794)( 10-15m
e ../4ii7'& = 0.302822
go/sin Ow
=
=
1.78266 x 10-l0 kg
2.99792 x 10-9esu
0.6295;
g",1 cosO", = 0.7180
28.760
(sin20
1.214
Conversion Factors
IA
I MeVjCl
-m,eluh,2 = -13.6057/"2eV
I'
Introduction
Elementary Particle Physics
Elementary particle physics addresses the question, 'What is matter made of?' at
the most fundamental level - which is to say, on the smallest scale of size. It's
a remarkable fact that matter at the subatomic level consists of tiny chunks, with
vast empty spaces in between. Even more remarkable, these tiny chunks come
in a small number of different
neutrinos. and so on), which are then replicated in astronomical quantities to make
all the 'stuff' around us. And these replicas are absolutely perfe<:t copies - not
just 'pretty similar', like two Fords coming off the same assembly line, but utterly
indistirlg\'ishabk You can't stamp an identification number on an electron, or paint
a spot on it - if you've seen one, you've seen them all. This quality of absolute
identicalness has no analog in the macroscopic world. (In quantum mechanics it
is reflected in the Pauli exclusion principle.) It enormously simplifies the task of
elementary particle physics: we don't have to worry about big electrons and little
ones, or new electrons and old ones - an electron is an electron is an electron. It
didn't have to be so easy.
My first job, then, is to introduce you to the various kinds ofelementary particles­
the actors, if you will, in the drama. I could simply list them, and tell you their
properties (mass, electric charge, spin, etc.), but I think it is better in this case
to adopt a historical perspective, and explain how each particle first came on the
scene. This will serve to endow them with character and personality, making them
easier to remember and more interesting to watch. Moreover, some of the stories
are delightful in their own right.
Once the particles have been introduced, in Chapter 1, the issue becomes, 'How
do they interact with one another?' This question, directly or indirectly, will occupy
us for the rest of the book. If you were dealing with two macroscopic objects, and
you wanted to know how they interact, you would probably begin by holding them
at various separation distances and measuring the force between them. That's how
Coulomb determined the law ofelectrical repulsion between two charged pith balls,
and how Cavendish measured the gravitational attraction of two lead weights. But
you can't pick up a proton with tweezers or tie an electron onto the end ofa piece of
string: they're just too small. For practical reasons, therefore, we have to resort to
In''''''''''''''" to El,m<>Wuy Pgrt;a..l. S<C<>nd Edj,jon. David Griffiths
CopyrightC 2008 W1LEY.vCH V�rl.1g GmbH &. Co. KG". Weinh�im
[SON: 978·)'s2HOCiOl·2
2!,ntroduaion
less direct means to probe the interactions of elementary particles. As it turns out,
almost all of our experimental information comes from three sources: (I) scattering
events. in which we fire one particle at another and record (for instance) the angle
of deflection; (2) decays. in which a particle spontaneously disintegrates and we
examine the debris; and (3) bound states. in which two or more particles stick
together. and we study the properties of the composite object. Needless to say,
determining the interaction law from such indirect evidence is not a trivial task
Ordinarily. the procedure is to guess a form for the interaction and compare the
resulting theoretical predictions with the experimental data,
The formulation of such a guess ('model' is a more respectable term for it) is
guided by certain general principles. in particular. special relativity and quantum
mechanics. In the diagram below I have sketched out four realms of mechanics:
Small_
Fast J.
Classical
Quantum
mechanics
mechanics
Relativistic
Quantum
mechanics
field theory
The world of everyday life, of course, is governed by classical mechanics. But for
objects that travel very fast (at speeds comparable to c). the classical rules are
modified by special relativity. and for objects that are very small (comparable to the
size of atoms. roughly speaking), classical mechanics is superseded by quantum
mechanics. Finally, for things that are both fast
and small, we require a theory
that incorporates relativity and quantum principles: quantum field theory. Now,
elementary particles are extremely small, of course. and typically they are also very
fast. So. elementary particle physics naturally falls under the dominion of quantum
field theory.
Please observe the distinction here between a typt: ofmechanics and a particumr
force mw. Newton's law of universal gravitation. for example. describes a specific
interaction (gravity), whereas Newton's three laws of motion define a mechanical
system (classical mechanics). which (within its jurisdiction) governs all interactions.
The force law tells you what F is, in the case at hand; the me<:hanics tells you how
to use F to determine the motion, The goal of elementary particle dynamics. then,
is to guess a set of force laws which. within the context of quantum field theory,
corre<:tiy describe particle behavior.
However. some general features of this behavior have nothi ng to do with the
detailed form of the interactions. Instead they follow directly from relativity, from
quantum mechanics. or from the combination of the two. For example. in relativity.
energy and momentum are always conserved, but (rest) mass is not. Thus the decay
6 _ P + If is perfe<:t1y acceptable, even though the 6 weighs more than the sum
of p plus If. Such a process would not be possible in classical mechanics, where
mass is strictly conserved. Moreover, relativity allows for particles of zero (rest)
mass - the very idea of a massless particle is nonsense in classical mechanics -
and as we shall see. photons and gluons
are massless,
fl�mt"Ul'l' Parode Physics
In quantum mechanics a physical system is described by its state, s (represented
by the wave function ..;, in SchrOdinger's fonnulation, or by the kit Is) in Dirac's
theory). A physical process, such as scattering or decay, consists of a transition
from one state to another. But in quantum mechanics the outcome is not uniquely
determined by the initial conditions; all we can hope to calculate, in general, is
the probability for a given transition to occur. This indeterminacy is reflected
the observed behavior of particles. For example, the charged pi meson ordinarily
disintegrates into a muon plus a neutrino, but occasionally one will decay into an
electron plus a neutrino. There's no difference in the original pi mesons; they're all
identical. It is simply a fact of nature that a given particle can go either way.
Finally, the union of relativity and quantum mechanics brings certain extra
dividends that neither one can offer by itself: the existence of antiparticles (with
the same mass and lifetime as the particle itself, but opposite ele<:tric charge), a
proof of the Pauli exclusion principle (which in nonrelativistic quantum mechanics
is simply an ad hoc hypothesis), and the so·called TCP thton:m. I'll tell you more
about these later on; my purpose in mentioning them here is to emphasize that
these are features of the mechanical system itself, not of the particular model.
Short of a catastrophic revolution, they are untouchable. By the way, quantum field
theory in aU its glory is difficult and deep, but don't be alanned: Feynman invented
a beautiful and intuitively satisfying formulation that is not hard to learn; we'll
come to that in Chapter 6. (The urivation of Feynman's rules from the underlying
quantum field theory is a different matter, which can easily consume the better
part of an advanced graduate course, but this need not concern us here.)
In the 1960s and 1970s a theory emerged that described all of the known
elementary particle interactions, except gravity. (As far as we can tell, gravity is
much too weak to play any significant role in ordinary particle processes.) This
theory - or, more accurately, this collection of related theories, based on two
families of elementary particles (quarks and leptons), and incorporating quantum
electrodynamics, the Glashow-Weinberg-Salam theory of electroweak processes,
and quantum chromodynamics - has come to be called the Standard MOOd. No
one pretends that it is the final word on the subject, but at least we are now playing
with a full deck of cards. Since 1978, when the Standard Model achieved the status
of 'orthodoxy', it has met every experimental test. Moreover. it has an attractive
aesthetic feature: all of the fundamental interactions derive from one general
principle, the requirement of local gaug& invarianc�. It seems certain that future
developments will involve extensions of the Standard Model, not its repudiation.
This book might be called an 'Introduction to the Standard Model'.
As that alternative title suggests, it is a book about elementary particle thtory,
with very little on experimental methods or instrumentation. These are important
matters, and an argument can be made for integrating them into a text such as this,
but they can also be distracting, and interfere with the clarity and elegance of the
theory itself. I encourage you to read about the experimental aspects of the subject,
and from time to time I will refer you to particularly accessible accounts. But for
now, I'll confine myself to scandalously brief answers to the two most obvious
experimental questions.
11
" 1,,,tmduaiQ,,
How Do You Produce Elemental)' Partides?
Electrons and protons are no problem; these are the stable constituents of ordinary
matter. To produce electrons one simply heats up a piece of metal, and they come
boiling off. If you want a beam of electrons, you just set up a positively charged plate
nearby, to attract them over, and poke a small hole in it; the electrons that make it
through the hole constitute the beam. Such an t/ectron gun is the starting element
in a television tube or an oscilloscope or an electron accelerator (Figure 1.1).
To obtain protons you ionize hydrogen (in other words, strip off the electron). In
fact, if you're using the protons as a targel, you don't even need to bother about the
electrons; they're so light that an energetic incident particle will knock them out of
the way. Thus, a tank of hydrogen is essentially a tank of protons. For more exotic
particles there are three main sources: cosmic rays, nuclear reactors, and particle
accelerators.
•
Cosmic rays: The earth is constantly bombarded with
high-energy particles (principally protons) coming from
outer space. What the source of these particles might be
remains something of a mystery; at any rate, when they hit
atoms in the upper atmosphere they produce showers of
secondary particles (mostly muons and neutrinos, by the
Fig. 1.1 SLAC; tht straight lint is tht acctltrator itstlf.
(Courttsy Stanford Untar Acctltratof Ctnttr.)
How Do You Produce Elementol)' P�rticles?
time they reach ground level), which rain down on us all the
time. As a source of elementary particles, cosmic rays have
two virtues: they are free, and their energies can be
enormous - far greater than we could possibly produce in
the laboratory. But they have two major disadvantages: the
rate at which they strike any detector of reasonable size is
very low, and they are completely uncontrollable. So cosmic
•
ray experiments call for patience and luck .
Nuclear r eactors: When a radioactive nucleus disintegrates, it
may emit a variety of particles - neutrons, neutrinos, and
what use<! to be called alpha rays (actually, alpha particles,
which are bound states of two neutrons plus two protons),
beta rays (actually, electrons or positrons), and gamma rays
•
(actually, photons).
Particle accelerarors: You start with electrons or protons,
accelerate them to high energy, and smash them into a target
(Figure 1.1). By skillful arrangements of absorbers and
magnets, you can separate the particle species that you wish
to study from the resulting debris. Nowadays it is possible in
this way to generate intense secondary beams of positrons,
muons, pions, kaons, B·mesons, antiprotons, and neutrinos,
which in tum can be fire<! at another target. The stable
particles - electrons, protons, positrons, and antiprotons can even be fed into giant norage rings in which, guided by
powerful magnets, they circulate at high speed for hours at a
time, to be extracted and used at the require<! moment [1J.
In general. the heavier the particle you want to produce, the higher must be
the energy of the collision. That's why, historically, lightweight particles tend to
be discovere<! first, and as time goes on, and accelerators become more powerful,
heavier and heavier particles are found. It turns out that you gain enormously in
rdativt energy if you collide two high-speed particles head-on, as oppose<! to firing
one particle at a stationary target. (Of course, this calls for much better aim!) Forthis
reason many contemporary experiments involve colliding beams from intersecting
storage rings; If the particles miss on the first pass, they can try again the next time
around. Indeed. with electrons and positrons (or protons and antiprotons) the samt
ring can be use<!, with the plus charges circulating in one direction and minus
charges the other. Unfortunately, when a charge<! particle accelerates it radiates.
thereby losing energy. In the case of circular motion (which, of course, involves
acceleration) this is calle<!
synchrotron radiatwn, and it severely limits the efficiency
of storage rings for energetic electrons (heavier particles with the same energy
accelerate less, so synchrotron radiation is not such a problem for them). For this
reason electron scattering experiments will increasingly tum to
linwr colliders,
while storage rings will continue to be used for protons and heavier particles.
I
S
'
I ,ntrodudion
There is another reason why particle physicists are always pushing fOf higher en·
ergies: in general, the higher the energy of the collision, the closer the two particles
come to one another. So if you want to study an interaction at very short range, you
nee<! very energetic particles. In quantum.mechanical terms, a particle of momen·
tum p has an associated wavelength A given by the de Broglie formula A
hlp,
where h is Planck's constant. At large wavelengths (low momenta) you can only
hope to resolve relatively large structures; in order to examine something extremely
small, you nee<! comparably short wavelengths, and hence high momenta. If you
ik
l e,
to make .6.:>: small, 6p must be large. However you look at it, the conclusion is the
same: to probe small distances you need high energits.
At present the most powerful accelerator in the world is the Twutro71 at Fermilab
(Figure 1.2), with a maximum beam energy of almost 1 TeV. The tevatron (a
proton-antiproton collide!) began operation in 1983; its successor, the Supercon.
ducting Supercollider (SSC) was under construction in 1993 when the project was
terminated by Congress. As a result, there has been a long period in which no
fundamental progress was possible. This dry spell should end in 2008, when the
Large Hadron Collider (LHC) at CERN starts taking data (Figure 1.3). The LHC is
designed to reach beam energies in excess of 7 TeV, and the hope is that this new
terrain will include the Higgs particle, possibly supersymmetric particles, and best of all - something completely unexpected [2J. It's not dear what comes after
the LHC - most likely the proposed International Unear Collider (ILC). But, accel·
erators have become so huge (the sse would have been 87 km in circumference)
that there is not much room for expansion. Perhaps we are approaching the end
"'"
Fig. 1.2 Fermilab; the large circle in the background is the
Tevatron. (Courtesy Fermilab Visual Media Services.)
HI)'" Do YI)" De�cI EI�m�nt"'y Plzrticles?
����-
FIg. 1.3 CERN; the c;,de indiates the pith of the LHC t�n·
nel (follllerly LEP) - Geneva and Mt Blanc are in the back·
ground. (Co�rtesy CERN.)
of the accelerator era, and particle physicists will have to tum to astrophysics and
cosmology for infonnation about higher energies. Or perhaps someone will have a
clever new idea for squeezing energy onto an elementary particle:
How Do You Deka Elementary Particles?
There are many kinds of particle dete<tors - Geiger counters, cloud chambers. bub·
ble chambers, spark chambers. drift chambers, photographic emulsions. Cerenkov
counters, scintillators, photomultipliers, and so on. Actually, a typical modem
•
In nw:roscopk terms the 3m<)unt of energy involved is not that ��t _ after �II. t TeV
(lOll .,V) is only 10-1 Jo�les; the problem is how 10 deli"", that energy tl> a p;irticle. NI> law
or physics prevents you from doing so, but nobody has yet figured out a way to do it without
gigantic (and �nslve) machinery.
17
'j ,ntrodUcJiOn
fig. 1.<4 Thf: COF df:tKtor at Ff:rmil;b, whf:rf: the top qua""
was diSCOVf:rW. (Courtesy Fermifab Visual MWia Services.)
dete<:tor has whole arrays of these devices, wire<! up to a computer that tracks
the particles and displays their traje<:tories on a television screen (Figure 1.4). lbe
details do not concern us, but there is one thing you should be aware of: most
dete<:tion mechanisms rely on the fact that when high-energy charge<! particles pass
through matter they ionize atoms along their path. The ions then act as 'see<ls' in
the formation of droplets (cloud chamber) or bubbles (bubble chamber) or sparks
(spark chamber), as the case may be. But electrically neutral particles do not cause
ionization, and they leave no tracks. For instance, ifyou look at the bubble chamber
photograph in Figure 1.9, you will see that the five neutral particles are 'invisible';
their paths have been re<:onstructe<! by analyzing the tracks ofthe charged particles
in the picture and invoking conservation of energy and momentum at each vertex.
Notice also that most of the tracks in the picture are cU"'td (actually, aU of them
are, to some extent; try holding a ruler up to one you think s
i straight). The bubble
particle ofcharge q and momentum p will move in a circle ofradius R given by the
chamber was place<! between the poles of a giant magnet; in a magnetic field B, a
famous cyclotronformula: R = pc/qB, where c is the speed of light. The curvature
of the track n
i a known magnetic field thus affords a very simple measure of the
particle's momentum. Moreover, we can immediately tell the sign of the charge
from the direction of the curve.
Units
Units
Elementary particles are small, so for our purposes the normal me<hanical units grams, ergs, joules, and so on - are inconveniently large. Atomic physicists
introduce<! the electron 1I0lt - the energy acquire<! by an ele<tron when accelerate<!
through a potential difference of 1 volt: 1 eV = 1.6 10-19 joules. For us the eV is
inconveniently small, but we're stuck with it. Nuclear physicists use keY (10l eV);
x
typical energies in particle physics are MeV (loG eV), GeV (109 eV), or even TeV
(1012 eV). Momenta are measure<! in MeV!c (or GeV!c, or whatever), and masses
in MeV1c2. Thus the proton weighs 938 MeV/c2 = 1.67 x 10-2-4 g.
Actually, particle theorists are lazy (or clever, depending on yourpoint ofview) they seldom include the c's and Ii's (Ii 5: h/2Jr) in their formulas. You're just
supposed to fit them in for yourselfat the end, to make the dimensions come out
right. As they say in the business, 'set c = Ii = 1'. This amounts to working in
units such that time s
i measure<! in centimeters and mass and energy in inverse
centimeters; the unit of time is the time it takes light to travel 1 em, and the
unit of energy is the energy of a photon whose wavelength is 211" em. Only at the
end of the problem do we revert to conventional units. This makes everything
look very elegant, but I thought it would be wiser in this book to keep al! the c's
and Ii's where they belong, so that you can che<k for dimensional consistency as
you go along. (If this offends you, remember that it is easier for you to ignore
an Ii you don't like than for someone else to conjure one up in just the right
place.)
Finally, there is the question of what units to use for ele<tric charge. In
introductory physics courses most instructors favorthe 51 system, in which charge
is measure<! in coulombs, and Coulomb's law reads
F
1_ ql q2
- _
-
41UO
r2
(51)
i done in the Gaussian system. in which charge is measure<!
Most advanced work s
in ekctrostatic units (esu), and Coulomb's law is written
F = q�2
(G)
But elementary particle physicists prefer the Hwvisilk
Coulomb's law takes the form
(HL)
The three units ofcharge are related as follows:
-
Lorentz system, in which
19
"I inlrodUG,;on
In this book I shall use Gaussian units exclusively, in order to avoid unnecessary
confusion n
i an already difficult subject. Whenever possible I wi!! express results
in terms of theftnl structure "'TlStant
�
ct = - = ---
tIC
1
137.036
where t is the charge of the electron in Gaussian units. Most elementary particle
texts write this as ;' /411". because they are measuring charge in Heaviside-Lorentz
units and setting c Ii 1; but everyone agrees that the number is 1/137.
= =
Further reading
Since the early 1%05. the Particle Data Group at Berkeley has periodically issued a
listing of the established particles and their properties. These are published every
other year in Reviews of Modern Physics or Jourmil of Physics G, and summarized
in a (free) booklet that can be ordered on the web at http:\\pdg.lbl.gov. In the early
days this summary took the fonn of 'wallet cards', but by 2(H)6 it had grown to a
densely packed 315 pages. ! shall refer to it as the Particle Physics Booklet (PPS).
Every student of elementary particle physics must have a copy - don't leave home
without it! The longer version, called the Review oJPartick Physics (RPP) is the bible
forprofessionals - the 2006 edition runs to 1231 pages, and it includes authoritative
articles on every relevant subject, written by the world's leading experts (3J. If you
want the definitive, up-to--clate word on any particular topic, this is the place to go
(it is also available on·line, at the Particle Data Group web site).
Particle phySiCS is an enormous and rapidly changing subject. My aim in this
book is to introduce you to some important ideas and methods, to give you a
sense of what's out there to be learned, and perhaps to stimulate your appetite
for more, If you want to read further in quantum field theory, I particularly
recommend:
Bjorken, J. D. and Drell, S. D. (1964) Relativistic Ql«lntum Mechanics and
Relativistic Quantum Fields, McGraw-Hill, New York.
Itzykson, C. and Zuber, ).·B. (1980) Quantum Field Theory, McGraw-Hi!!, New
York.
Peskin. M. E. and Schroeder, O. y, (1995) An Introduction to Quantum Field
Theory, Perseus, Cambridge, M.A.
Ryder, L. H. (1 985) Quantum Field Theory, Cambridge University Press,
Cambridge, UK.
Sakurai, J. J. (1967) Advanud Quantum Mechanics, Addison-Wesley, Reading,
M.A.
! warn you, however, that these are all difficult and advanced books.
Furth" fUldjng
For elementary particle physics itself, the following books nisted in order of
increasing difficulty) are especially useful:
Close, F., M arten M. and Sutton, C. (1987) Tht Particle Explosion, Oxford
University Press, Oxford, UK.
Frauenfelder, H. and Henley, E. M. (1991) Subatomic Physics, 2nd edn,
Prentice·Hall. Englewood Cliffs, N.J.
Gottfried, K. and Weisskopf, V. F. (1984) Concepts of Particle Physics, Oxford
University Press. Oxford.
Perkins, D. H. (2000) Introduction 10 High.Energy Physics. 4th Ed, Cambridge
Univ ersity Press, Cambridge, UK.
,
Halzen, F. and Martin, A. D. (1984) Quarks and Leptons, John Wiley & Sons,
ltd, New York.
Roe, B. P. (1996) Particle Physics at the New Millennium, Springer, New York.
Aitchison, !. J. R. and Hey, A. J. G. (2003) Gauge Thrones in Particle Physics,
3rd edn, Institute of PhySiCS, Bristol, UK.
Seiden, A. (2005) Particle Physics: A Comprehensive Introduction, Addison­
Wesley, San Francisco, c.A.
Quigg, C. (1997) Gauge Throrits of the Strong. Weak, and Eltclromagnetic
inJemctions, Addison-Wesley, Reilding. M.A.
References
1 For � comprehensive bibliogra­
phy � Chao, Q. W. (2006) AinU­
jean JournIll of Phyoic.!, 1.. 855.
2 Smith, C. L (July 2000) Soi.ntific
A_ricon. 11: (a) lederman, L
(2001) Noture, "48. 310.
1 The current reference is Yao. W.-M.
a al (2006) Journal of Physic.!. G 33.
.
I. But because you will want to
u� the most re<:ent versions,
Partick PhysiCl Bookld and R.­
view cif Particle Physicl, appt:nd·
ing the year when it is relevllnt.
I will simply rerer to them �s
I II
I"
,
Historical Introduction to the Elementary Particles
This chaptu is a kind of yolk history' of dtmentary particle physic.s, Its purpQSe is
to provide a sense of how the various particie5 were first discovered, and how thty fit
into the overall scheme oj things. Along the way some oj the fundamental ideas that
dominau elementary particle thtcry art txplaintd. This mattrial should be read quickly,
as background to the rt5! o
flhe book. (As hiswry, the picture preunted here is cutai,lly
misleading,Jor it sticks clostly to the main track, ignoring 1mfo1se starts and blind alleys
that accompany the devdopment ofany seiena. That's why I caU it Jolk' history - it's
the wayparticlephysicists like to remember the subject - a sucussion a/brilliant insights
and heroic triumphs unmarred byfoolish mistakes, conjUsion, andfrustration. It wasn't
rtaUy quitt so easy.)
, .,
The Classical Era (1897-1932)
It is a little artificial to pinpoint such things, but I'd say that elementary particle
physics was born in 1897, with J. J. Thomson's discovery of the electron [1[. (It
is fashionable to carry the story all the way back to Democritus and the Greek
atomists, but apart from a few suggestive words their metaphysical speculations
have nothing in common with modern science, and although they may be of
modest antiquarian interest, their genuine relevance is negligible.) Thomson knew
that catholk rays emitted by a hot filament could be deflected by a magnet. This
suggested that they carried electric charge; in fact, the direcion
t
of the curvature
required that the charge be negative. It seemed, therefore, that these were not rays
at all, but rather streams of particles. By passing the beam through crossed electric
and magnetic fields, and adjusting the field strength until the net deflection was
zero, Thomson was able to detennine the velOCity of the particles (about a tenth the
speed oflight) as well as their charge-to-mass ratio (Problem 1.1). This ratio turned
out to be enonnously greater than for any known ion, indicating either that the
charge was extremely large or the mass was very small. Indirect evidence pointed
to the second conclusion. Thomson called the particles corpU5cks. Back in 1891.
George Johnstone Stoney had introduced the term 'electron' for the fundamental
unit of charge; later, that name was taken over for the particles themselves.
In,rcd"",km 10 s."",tIl/Zry Parlicks. StwnJ Edi'km. Oa,,;d Griffiths
Copyright C 2008 W!lEY·VCH V�rllg GmbH &Co. KGaA. W.inh.im
!SBN: 978·)·527·40601·2
'''1 J Hi,torial/ lntroduction to the EI�mMta..,. Purfid�,
Thomson correctly surmised that these electrons were essential constituents
of atoms; however, since atoms as a whole are electrically neutral and very
much heavier than electrons, there immediately arose the problem of how the
compensating plus charge � and the bulk of the mass � is distributed within an
atom. Thomson himself imagined that the electrons were suspended in a heavy,
positively charged paste, like (as he put it) the plums in a pudding. But Thomson's
model was decisively repudiated by Rutherford's famous scattering experiment,
which showed that the positive charge, and most ofthe mass, was concentrated in
a tiny core, or nuc/tus, at the center of the atom. Rutherford demonstrated this by
firing a beam of a particles (ionized helium atoms) into a thin sheet of gold foil
(Figure 1.1). Had the gold atoms consisted of rather diffuse spheres, as Thomson
supposed, then all of the a particles should have been deflected a bit, but none
would have been deflected much � any more than a bullet is deflected much when
it passes, say, through a bag of sawdust. What infacl occurred was that mOSI of
the a particles passed through the gold completely undisturbed, but a few of them
bounced off at wild angles. Rutherford's conclusion was that the a particles had
Zinc sulfide screen
Gold foil
Collimated beam
of a--particles
Source of
a·particles
Vacuum
pump
fig. 1.1 SchematiC diagram of the ilPPilratus used in the
Rutherford scanering experiment. Alpha pilrtides scattered
by the gold foil strike a fluorescent screen, giving off a flilsh
of light, which ;5 observed visually through a microscope.
1.2 Th� Photon (1900-1924)
encountered something very small, very hard, and very heavy. Evidently the positive
charge, and virtually all ofthe mass, was concentrated at the center, occupying only
a tiny fraction ofthe volume ofthe atom (the electrons are too light to play any role in
the scattering; they are knocked right outofthe way bythe much heavier [t particles).
The nucleus of the lightest atom (hydrogen) was given the name protol1 by
Rutherford. In 1914 Niels Bohr proposed a model for hydrogen consisting of a
single electron circling the proton, rather like a planet going around the sun, held
in orbit by the mutual attraction of opposite charges. Using a primitive version of
the quantum theory, Bohr was able to calculate the spectrum of hydrogen, and the
agreement with experiment was nothing short of spectacular. It was natural then
to suppose that the nuclei of heavier atoms were composed of two or more protons
bound together, supporting a like number oforbiting electrons. Unfortunately, the
next heavier atom (helium), although it does indeed cany two electrons, weighsfour
times as much as hydrogen, and lithium (three electrons) is stvel1 times the weight of
hydrogen, and so it goes. This dilemma was finally resolved in 1932 with Chadwick's
discovery of the neutron - an electrically neutral twin to the proton. The helium
nucleus, it turns out, contains two neutrons in addition to the two protons; lithium
evidently includes four; and, in general, the heavier nuclei carry very roughly the
same number ofneutrons as protons. (The number ofneutrons is in fact somewhat
flexible - the same atom, chemically speaking, may come in several different iso­
topes, all with the same number ofprotons, but with varying numbers ofneutrons.)
The discovery ofthe neutronput the final touch on what we might call the classical
�riod in elementary particle physics. Never before (and I'm sorry to say never since)
has physics offered so simple and satisfying an answer to the question, 'What is
matter made of?' In 1932, it was all just protons, neutrons, and electrons. But
already the seeds were planted for the three great ideas that were to dominate the
middle �riod (1930-1960) in particle physics: Yukawa's meson, Dirac's positron,
and Pauli's neutrino. Before we come to that, however, I must back up for a
moment to introduce the photon.
1.2
The Photon (1900-1924)
In some respects, the photon s
i a very 'modem' particle, having more in common
with the W and Z (which were not discovered until 1983) than with the classical
trio. Moreover, it's hard to say exactly when or by whom the photon was really
'discovered', although the essential stages in the process are clear enough. The
first contribution was made by Planck in 1900. Planck was attempting to explain
the so-called blackbody sp«trum for the electromagnetic radiation emitted by
a hot object. Statistical mechanics, which had proved brilliantly successful in
explaining other thermal processes, yielded nonsensical results when applied to
electromagnetic fields. In particular, it led to the famous 'ultraviolet catastrophe',
pre<licting that the total power radiated should be Il1fil1itt. Planck found that he could
escape the ultraviolet catastrophe - and fit the eKpt'rimental curve - ifhe assumed
115
16 1 ! Hi�ori,ol lnfroducliM
fa the fkmenfory Par/ides
that electromagnetic radiation is quantized, coming in little 'packages' of energy
E "" hv
(1.1)
where v is the frequency ofthe radiation and h is a constant. which Planck adjusted
to fit the data. The modern value of Planck's constant is
h "" 6.626 )( 10-17 erg s
(1.2)
Planck did not profess to know why the radiation was quantized; he assumed that
it was due to a peculiarity in the emission process: for some reason a hot surface
only gives off light" in little squirts.
Einstein, n
i 1905, put fOIWard a farmore radicalview. Heargued that quantization
was a feature of the electromagnetic field itself, having nothing to do with the
emission mechanism. With this new twist, Einstein adapted Planck's idea, and his
fonnula, to elCpiain the phcl«.lulric effect: when electromagnetic radiation strikes a
metal surface, electrons come popping out. Einstein suggested that an incoming
light quantum hits an electron in the metal. giving up its energy (hv); the excited
electron then breaks through the metal sunace. losing in the process an energy w
(the so·called work functwn of the material - an empirical constant that depends
on the particular metal involved). The electron thus emerges with an energy
E ::; hv - w
(1.3)
(It may lose some energy before reaching the surface; that's the reason for the
inequality.) Einstein's formula (Equation 1.3) is trivial to derive, but it carries an
extraordinary implication: The maximum electron energy is indeptndent of the
intensity of the light and depends only on its color (frequency). To be sure, a more
intense beam will knock out mort electrons, but their enagies will be the same.
Unlike Planck's theory, Einstein's met a hostile reception, and over the next
20 years he was to wage a lonely battle for the light quantum [2). In saying that
electromagnetic radiation is by its nature quantized, regardless of the emission
mechanism, Einstein came dangerously dose to resurrecting the discredited parti.
de theory oflight. Newton, of course, had introduced such a corpuscular model, but
a major achievement of nineteenth-century physics was the decisive repudiation
of Newton's idea in favor of the rival wave theory. No one was prepared to see
that accomplishment called into question, even when the experiments came down
on Einstein's side. In 1916 Millikan completed an exhaustive study of the photo·
electric effect and was obliged to report that 'Einstein's photoeleo:tric equation .
appears in every case to predict exactly the observed results. . . . Yet the semicor·
puscular theory by which Einstein arrived at his equation seems at present wholly
untenable' [lJ.
• In this book the W(lrd lighl sunds for declroml>gPltlic ",di�lion. whether or not it happens to fall
in the visible "'gion.
1.2 The Pho!<Jn (I900-1924j
�
.... (X)
o
m
-_£_
m� - ­
Before
Alter
Fig. 1.2 Compton scattering. A photon of wavelength ).. scat·
ters off a �rticle, initi311� 3t ,est, of mass m. The scattertd
photon carries wavelength ).' given b� Equation 1.4.
What finally settled the issue was an experiment conducted by A. H. Compton
in 1923. Compton found that the light scattered from a particle at rest is shifted in
wavelength. according to the equation
(1.4)
},' = ).. + ,1,.« I - cosO)
where ).. is the inddent wavelength, ).,' is the scattered wavelength. B is the scattering
angle, and
A, = h/mc
(1.5)
is the so-called Compto" wavekngth of the target particle (mass mI. Now, this is
pruisely the formula you get (Problem 3.27) if you Ireat light as a particle of zero
-
rest mass with energy given by Planck's equation, and appJy the laws of conser·
just as you would for an ordinary
elastic collision (Figure 1.2). That clinched it; here was direct and incontrovertible
vation of (relativistic) energy and momentum
experimental evidence that light behaves as a particle, on the subatomic scale. We
call this particle the photon (a name suggested by the chemist Gilbert Lewis, in
1926); the symbol for a photon s
i y (from gamma ray). How the particle nature of
light on this level is to be reconciled with its well·established wave behavior on the
macroscopic scale (exhibited in the phenomena of interference and diffraction) is
a story I'll leave for books on quantum mechanics.
Although the photon initiallyforud itselfon an unreceptive community ofphysi·
cists, it eventually found a natural place in quantum field theory, and was to offer a
whole new perspective on electromagnetic interactions. In classical electrodynam.
ics, we attribute the electrical repulsion of two electrons, say, to the electric field
surrounding them; each electron contributes to the field, and each one responds to
the field. But in quantum field theory, the electric field is quanlized (in the form of
photons), and we may picture the interaction as consisting of a stream of photons
passing back and forth between thetwo charges, each electron continually emitting
photons and continually absorbing them. And the same goes for any noncontact
1 17
18
I 1 Historicol lnlrodllccion the Element",..,. Pr1t1icles
10
force: Where classically we interpret 'action at a distance' as 'mediated' by afield,
we now saythat it is mediated by an txchangt ofpanicles (the quanta ofthe field). In
the case of electrodynamics, the mediator is the photon; for gravity, it is called the
graviton (though a fully successful quantum theory ofgravity has yet to be developed
and it may well be centuries before anyone detects a graviton experimentally),
You will see later on how these ideas are implemented in practice, but for now
I want to dispel one common misapprehension. When I say that every force is
mediated by the exchange of particles, I am /Wt speaking of a merely killtmatic
phenomenon. Two ice skaters throwing snowballs back and forth wi
l l of course
move apart with the succession of recoils; they 'repel one another by exchange of
snowballs', if you like. But that's not what s
i involved here. For one thing, this
mechanism would have a hard time accounting for an attractivt force. You might
think of the mediating particles, rather, as 'messengers', and the message can just
as well be 'come a little closer' as 'go away'.
I said earlier that in the 'classical' picture ordinary matter is made of atoms,
in which electrons are held in orbit around a nucleus of protons and neutrons
by the electrical attraction of opposite charges. We can now give this model a
more sophisticated formulation by attributing the binding force to the exchange
of photons between the electrons and the protons n
i the nucleus. However, for
the purposes of atomic physics this is overkill, for in this context quantization ofthe
electromagnetic field produces only minute effects (notably the Lamb shift and the
anomalous magnetic moment of the electron). To excellent approximation we can
pretend that the forces are given by Coulomb's law (together with various magnetic
dipole couplings). The point s
i that in a bound state enormous numbers ofphotons
are cont
inually streaming back and forth, so that the 'lumpiness' of the field is
effectively smoothed out, and classical electrodynamics is a suitable approximation
to the truth, But in most elementary particle processes, such as the photoelectric
effect or Compton scattering, individual photons are involved, and quantization can
no longer be ignored.
1.3
Mesons (1934-19-i7)
Now there is one conspicuous problem to which the 'classical' model does not
address itselfat all: what holds the nuckus together? After all, the positively charged
protons should repel one another violently, packed together as they are in such close
proximity, Evidently there must be some other force, more powerful than the force
of electrical repulsion, that binds the protons (and neutrons) together; physicists of
that less imaginative age called it, simply, the stron gforu. But if there exists such
a potent force n
i nature, why don't we notice it in everyday life? Thefact is that
virtually every force we experience directly, from the contraction ofa muscle to the
explosion of dynamite, is electromagnetic in origin; the only exception, outside a
nuclear reactor or an atomic bomb, is gravity. The answer must be that, powerful
though it is, the strong force is of very short ra"&t. (The range of a force is like the
1.3 Mesons ()934-1947)
arm's reach of a boxer - beyond that distance its influence falls off rapidly to zero.
Gravitational and electromagnetic forces have in.fini� range, but the range of the
strong force is about the size of the nucleus itself.)"
The first significant theory of the strong force was proposed by Yukawa in 1934
[4]. Yukawa assumed that the proton and neutron are attracted to one another by
some sort of jitld, just as the electron is attrilcted to the nucleus by an electric
field and the moon to the earth by a gravitational field. This field should properly
be quantized, and Yukawa asked the question: what must be the properties of its
quantum
-
the particle (analogous to the photon) whose exchange would account
for the known features ofthe strong force? For example, the short rilnge ofthe force
indicated that the mediator would be rather heavy; YUkawil calculated thilt its mass
should be nearly 300 times that ofthe electron, orilbout iI sixth the mass ofa proton
(see Problem 1.2). Because it fell between the electron and the proton, Yukawa's
particle came to be known as the mtSon (meaning 'middle·weigh!'). In the same
spirit, the electron is called a lepton ('light-weight'), whereas the proton and neutron
are baryons (,heavy-weight'). Now, Yukawa knew that no such particle had ever
been observed in the laboratory, and he therefore assumed his theory was wrong.
But at that time a number of systematic studies of cosmic rays were in progress,
and by 1937 two separate groups (Anderson and Ne.::!denneyer on the West Coast,
and Street and Stevenson on the East) had identified p<lrticles matching Yukawa's
description.t Indeed, the cosmic rays with which you are being bombarded every
few seconds as you read this consist primarily ofjust such middle-weight particles.
For a while everything seemed to be in order. But as more detailed studies
of the cosmic ray particles were undertaken, disturbing discrepancies began to
appear. They had the wrong lifetime and they seemed to be significantly lighter
than Yukawa had pre.::!icted: worse still, different ma.ss measurements were not
consistent with one another. In 1946 (after a period n
i which physicists were
engaged in a less savory business) decisive experiments were carried out n
i Rome
demonstrating that the (osmic ray particles interacted very weakly with atomic
nuclei [5]. If this was really Yukawa's meson, the transmitter of the strong force,
the interaction should have been dramatic. The puzzle WilS finally resolved in 1947,
when Powell and his coworkers ilt Bristol [6} discovered thilt there are actually
two middle-weight particles in (osmic rays, which they called If (or 'pion') and It
grounds [7].) The true Yukawa meson is the n; it is produced copiously in the upper
(or 'muon'). (Marshak reached the same conclusion Simultaneously, on theoretical
atmosphere, but ordinarily disintegrates long before reaching the ground (see
Problem 3.4). Powell's group expose.::! their photographiC emulsions on mountlin
tops (see Figure 1.3). One ofthe decay products is the lighter (and longer lived) It,
and it is primarily muons that one ohselVes at sea level. In the search for Yukawa·s
meson, then, the muon was simply an m
i postor, having nothing whatever to do
• This s
i a bit of an oversimplification. Typically. the fora-s go like ,-<'IOI/rl. where � is the
·rang.-' For Coulomb's law and N.-wton·s law ofunive=! gravitation. a '" 00: for the strong
fOICe a is about iO-ll em (I fml.
t Actwlly. it was Rob:.rt Op""nheimer who d� the connection betwttn these os i ray p,nti·
des and Yukawa·s me<an.
c mc
1 19
20
I J Hirtorir;ollmroductiol1
"
.l
10 Ih� fl�m�l1rory Portid�,
!,
\
\.�
"-
(
I » .- . - --
m·
,
--.
Fig. 1.) One of Powell"s earliest pictures
showing the track of iI pion in a photo·
trad). (50Il"'�: Powell. C. F•. Fowler, P. H.
and Perkins. D. H. (1959) The Study of fie·
graphic emulsion exposed to cosmic rays at
high altit ude. The pion (entering from the
men"',), Particles by 1m Phol(}grop/lic Merhod
Pergamon, New York. First publ ished in
left) decays into a muon ind a neutrino (the
(19-47) NaMe 159, 69-4.)
latter ;s electrically neutral, and leaves no
wi ththe strong in terac tio n s. In f
a ct , ti beha vesin every wa ylik ea hea vi er version
o fthe el ectron an dpro perl ybelon gs in the Itpkm famil y (thoughsom e peopl e to
this da y call it the 'mu-meson ' byfo rc eo fhabit) .
1.4
Antiparticles (1930-1956)
Nonr e!a tivistic quantum m echan c
i s was compl etedin theas to nishin gl ybri efp e­
riod 1923-1926,but the r ela ti vis tic version p ro ved to bea muc htho rn ier p robl em .
1.4 AntipDrticies (1930-1956)
The first major achievement was Dirac's discovery, in 1927, of the equation that
energy given by the relativistic formula E! - pZcz = mZ,4. But it had a very trou·
bears his name. The Dirac equation was supposed to describe free electrons with
bling feature: for every positive-energy solution (E = +Jp2c2 + m2c"') it admitted
=
_Jp2ci+ mZc"). This meant
a corresponding solution with negative energy (E
that, given the natural tendency of every system to evolve in the direction oflower
energy, the electron should 'runaway' to increasingly negative states, radiating
off an infinite amount of energy in the process. To rescue hs
i equation, Dirac
proposed a resolution that made up in brilliance for what it lacked in plausibility:
he postulated that the negative-energy states are all filled by an infinite 'sea' of
electrons. Because this sea is always there, and perfectly unifonn, it exerts no net
force on anything, andwe are not normally aware ofit. Dirac then invoked the Pauli
eJ((lusion principle (which says that no two electrons can occupythe same state), to
'explain' why the electrons we do observe are confined to the positive-energy states.
But if this is true, then what happens when we impart to one of the electrons in
the 'sea' an energy sufficient to knock it into a positive-energy state? The absence
of the 'expected' electron in the sea would be interpreted as a net positive charge
n
i that location, and the absence ofits expected negative energy would be seen as a
net positive energy. Thus a 'hole in the sea' would function as an ordinary particle
with positive energy and positive charge. Dirac at first hoped that these holes might
be protons, but it was soon apparent that they had to carry the same mass as the
ele<:tron itself - 2000 times too light to be a proton. No such particle was known
at the time, and Dirac's theory appeared to be in trouble. What may have seemed a
fatal defe<:t in 1930, however, turned n
i to a spectacular triumph in late 1931, with
Anderson's discovery of the positron (Figure 1.4), a positively charged twin for the
electron, with pre<:isely the attributes Dirac required [8J.
Sti!!, many physicists were uncomfortable with the notion that we are awash in
an infinite sea of invisible electrons, and in the 194()s Stuckelberg and Feynman
provided a much simpler andmore compelling interpretation ofthe negative-energy
states. In the Feynman-Stuckelberg formulation, the negative·energy solutions
are re-expressed as positive-energy states of a dilfrrent particle (the positron); the
electron and positron appear on an equal footing, and there is no need for Dirac's
'ele<:tron sea' or for its mysterious 'holes'. We'll see in Chapter 7 how this - the
modern interpretation - works. Meantime, it turned out that the dualism in Dirac's
equati
kind of particle there must exist a corresponding antiparticle, with the same mass
but opposite electric charge. The positron, then, is the antieltetron. (Actually, it is
i h the
in principle completely arbitrary which one you call the 'particle' and whc
'antiparticle' - I could just as well have said that the electron is the antipositron.
But since there are a lot of electrons around. and not so many positrons, we tend to
think ofelectrons as 'matter' and positrons as 'antimatter'). The (negativelycharged)
antiproton was first observed experimentally at the Berkeley Bevatron in 1955, and
the (neutral) antineutron was discovered at the same facility the following year [9J.
1 21
221 1 HisloriUll lnlroduclkm to the f/,montary PartiC/,s
/.<
Fi,. 1.4 The positron. In 1932. Ande�on
took this photograph of th", track l...ft in
a cloud chamb...r by a cosmic ray particle.
Th", ci1amb",r WiS plac...d in a magnetic field
(pointing into th", page), which caus",d th",
particle to travel in a <curve. But was it a
negatiye charge trilv...ling dOWl1ward or a
positi"", charg'" trayeling upward? In order
to distinguish. Anderson had placed a lead
pla.te across the center of the chimber (the
thid hori�ontal 1ine in ft.e photograph). A
particle passing through the plate slows
down, and subsequently moyes in it tighter
circle. By inspection of the curves. it is clear
that this particle tra"",l...d upward, and hence
.
must hive been positively charg...d From the
curvature of the track and from its texture.
Andenoon WiS able to show that th'" mass
of the particle was close to that of the ele<:­
Technology.)
tron. (Photo courtesy California Institute of
The standard notation for antiparticles is an overbar. For example, p denotes the
proton and p the antiproton; n the neutron and ii the antineutron. However, in
some cases it is customary simply to spedfy the charge. Thus most people write e+
for the positron (nott) and p..+ for the antimuon (not jl).' Some neutral particles are
their own antiparticles. For example, the photon: V iii y. In fact, you may have been
wondering how the antineutron differs physically from the neutron, since both are
uncharged. The answer is that neutrons carry other 'quantum numbers' besides
charge (in particular, baryon number), which change sign for the antiparticle.
Moreover, although its ntt charge s
i zero, the neutron does have a charge structure
(positive at the center and neaT the surface, negative in between) and a magnetic
dipole moment. These, too, have the opposite sign for ii.
There is a general principle in particle physics that goes under the name of
crossing symmttry. Suppose that a reaction ofthe fonn
is known to occur. Any of these particles can be 'crossed' over to the other side of
the equation, provided it is turned into its antiparticle, and the resulting interaction
, But you must not mix conventions: � s
i ambiguous. like a double negative - the reader d�n'l
know if you mean the positron or the ..nlipositron, (which is to say, the ekctron).
1.5 Neufrinos (1930-1962)
will also be allowed. For example.
A --+ B + C + D
A + C --+ B + D
C + D ..... A + B
In addition, the rtvtrst reaction occurs: C + D --+ A + B, but technically this
derives from the principle of dttaikd balanct, rather than from crossing symmetry.
Indee<l, as we shall see, the calculations involved in these various reactions are
practically identicaL We might almost regard them as different manifestations
of the same fundamental process. However, there is one important cavwl in all
this: conselVation of energy may veto a reaction that is otherwise permissible.
For example, if A weighs less than the sum of B, C, and D, then the decay
A --+ B + C + D cannot occur; similarly, if A and C are light, whereas B and D
are heavy, then the reaction A + C ..... B + D will not take place unless the initial
kinetic energy exceeds a certain 'threshold' value. So perhaps I should say that the
crossed (or reversed) reaction is dynamically permissible, but it may or may not be
kinematically allowed. The power and beauty of crossing symmetry can scarcely be
exaggerated. It tells us, for instance, that Compton scattering
is 'really' the same process as pair annihilation
although in the laboratory they are completely different phenomena.
The union of spe<ial relativity and quantum mechanics, then, leads to a pleasing
matter/antimatter symmetry. But this raises a disturbing question: how come our
world is populated with protons, neutrons, and ele<trons. instead of antiprotons,
antineutrons, and positrons? Matter and antimatter cannot coexist for long - if a
particle meets its antiparticle, they annihilate. So maybe it's just a historical accident
that in our corner ofthe universe therehappened to be more matter than antimatter,
and pair annihilation has vacuumed up all but a leftover residue of matter. If this
is so, then presumably there are other regions of space in which antimatter
predominates. Unfortunately, the astronomical evidence is pretty compelling that
all of the obselVable universe is made of ordinary matter. In Chapter 12 we will
explore some contemporary ideas about the 'matter-antimatter asymmetry'.
1.5
Neutrinos (1930-1962)
For the third strand in the story we return again to the year 1930 [10J. A problem
had arisen in the study ofnuclear beta de<ay. In beta decay, a radioactive nucleus A
[ 23
2-41
J HiJroriw! 'nlrodlJaion 10 the ElemenlQry Par1id�
is transfonned into a slightly lighter nucleus B, with the emission of an electron:
(1.6)
Conservation ofcharge requires that B carry one more unit of positive charge than
A. (We now realize that the underlying process here is the conversion ofa neutron,
in A, into a proton, in B; but remember that in 1930 the neutron had not yet
been discovered.) Thus the 'daughter' nucleus (B) lies one position farther along
on the periodic table. There are many examples of beta decay: potassium goes to
calcium (�K -+ ;:: Ca), copper goes to zinc (�Cu -+ � Zn), tritium goes to helium
ttH -+� He), and so on.'
Now, it is a characteristic of two·body decays
(A -+ B + q that the outgoing
energies are kinematically detennined, in the center-of·mass frame. Specifically,
if the 'parent' nucleus
(A) s
i at rest, so that B and e come out back·to-back with
equal and opposite momenta, then conservation ofenergy dictates that the electron
energy is (Problem 3.19)
(1.7)
The point to notice s
i that E isfixtd once the three masses are specified. But when
the experiments are done, it is found that the emitted electrons vary considerably in
energy; Equation 1.7 only determines the maximum electron energy for a particular
beta decay process (see Figure 1.5).
This was a most disturbing result. Niels Bohr (not for the first time) was ready to
abandon the law of conservation of energy.t Fortunately, Pauli took a more sober
view, suggesting that another particle was emitted along with the electron, a silent
accomplice that carries off the 'missing' energy. It had to be electrically neutral, to
conserve charge (and also, ofcourse, to explain why it left no track); Pauli proposed
to call it the �utron. The whole idea was greeted with some skepticism, and in
1932 Chadwick preempted the name. But in the follOwing year Fermi presented
a theory of bela decay that incorporated Pauli's particle and proved so brilliantly
successful that Pauli's suggestion had to be taken seriously. From the fact that the
observed electron energies range up to the value given in Equation 1.7 it follows
that the new particle must be extremely light; Fenni called it the neutrino ('little
neutral one'). For reasons you'll see in a moment, we now call it the antineutrino.
• The upper !lumber is the alOm;'; ....iglll (the
number of n�trons plus protons) �nd the
lower number is the atom;'; n..mber (the num·
ber of protons).
t [\ is interesting to note that Bohr was an
outspoR!I critic of Ensl6n's Ught qu�n.
I:Ilm (prior to 1924), that he mercilessly
denounced Schrodinger'1 eq�tion, dis·
(ouraged Dir�c's work on the rel�tivistic
e\e<tton th�ry (telling him, incorr«dy, that
](Je;n �nd Gordon hld llreldy SUC(eeded).
opposed Pauli's imroduction of th� n�utrino,
ridiculed Yukaw�'s th�ry of the meson, and
disparaged Feynman's approach 10 q�ntum
electrodynlmics. Great scientists do not al·
ways h:ove good judgment - especiaUy when
it concerns other people's work - but Bohr
mu.r hold the alJ·tim� r«ord.
1.S Newinos(19JO-1962)
�
,
•
1000
>
•
0
•
•
.�,
K
E,
500
8
..
0
z
•
5
Electron kinetic energy (KeV)
I.
Fig. 1.5 The beta decay �pectrum of tritium (/H
-+
15
2•
�H�).
(Soune: lewis, G. M. (1970) NeUln'ncs. Wykehim, London.
p. 30.)
In modern tenninology, then, the fundamental bela decay process is
(l.S)
(neutron goes to proton plus electron plus antineutrino).
Now, you may have notice<! something peculiar about Powell's picture of the
disintegrating pion (Figure 1.3): the muon emerges at about 90Q with respe<:t to
the original pion direction. (1bat's not the result ofa collision, by the way; collisions
with atoms in the emulsion account for the dither in the tracks, but they cannot
produce an abrupt left turn.) What this kink indicates is that some other particle
was produced in the de.::ay of the pion, a particle that left no footprints in the
emulsion, and hence must have been electrically neutral. It was natural (or at any
rate tccnQmica� to suppose that this was again Pauli's neutrino:
(1.9)
A few months after their first paper, Powell's group published an even more striking
picture, in which the subsequent decay of the muon is also visible (Figure 1.6). By
then muon decays had been studied for many years, and it was well established
that the charged secondary is an electron. From the figure there is dearly a neutral
product as well, and you might guess that it is another neutrino. However, this
time it is actually two neutrinos:
p. _ e + 2"
(1.10)
12S
261 1 His�ri�I I"lroduClio" 10
the Elementory PQrticies
•
1 ·r
,
I
!�
•
�
l_1C
-<
Fig. 1.6 Here. a pion deays into a muOn (plus . neutrino);
the muon subSe<:luently decays into an electron (and two
neutrinos). (Souru: Powell, C. F., Fowler, P. H. and Perkins,
o. H. (1959) � Study of Elementary Particle. by the Pho­
tographi, Method Pergamon, New York. First published in
(1949) NOlun 163, 82.)
How do we know there are two of them? Same way as before: we repeat the
experiment over and over, each time measuring the energy of the electron. If it
always comes out the same, we know there are just two particles in the final state.
But if it varies, then there must be (at least) three: By 1949 it was clear that the
• Here, and in the original beta decay prob­
lem. conservation of angular momentum
also requires a third outgoing partide, quite
indeptndently ofenergy conservation. But
the spin assignments were not so dear i n
the early days. and for most people ener8)'
con,",rvation Wi!; the compelling argument.
In the n
i terest of simplicity. I will keep
angular momenUlm out of the story until
Chapter •.
1.5 N�lJtri""'(19JO-I962)
electron energy in muon decay is not fixed, and the emission of two neutrinos was
the accepted explanation, (By contrast, the muon energy in pion decay is perf"ecdy
constant, within experimental uncertainties. confirming that this is a genuine
two-bo£Iy decay.)
By 1950, then, there was compeUing theoretical evidence for the existence
of neutrinos, but there was still no direct expenmeJ1tal verification. A skeptic
might have argued that the neutrino was nothing but a bookkeeping device - a
purely hypothetical particle whose only function was to rescue the conservation
laws. It len no tracks. and it didn't decay; in fact, no one had ever seen a
neutrino do anything. The reason for this is that neutrinos interact extraordinarily
weakly with matter; a neutrino of moderate energy could easily penetrate a
thousand light years(!) of lead.' To have a chance of detecting one you need
an extremely intense source. The decisive experiments were conducted at the
Savannah River nuclear reactor in South Carolina, in the mid-1950s. Here Cowan
and Reines set up a large tank of water and watched for the 'inverse' beta decay
reaction
(1.11)
At their detector the antineutrino Aux was calculated to be 5 X lOll particles per
square centimeter per second. but even at this fantastic intensity they could only
hope for two or three events every hour- On the other hand, they developed an
ingenious method for identifying the outgoing positron. Their results provided
unambiguous confirmation ofthe neutrino's existence [11J.
As I mentioned earlier, the particle produced in ordinary beta decay is actually
an antineutrino, not a neutrino. Of course, since they're electrically neutral,
you might ask - and many people did - whether there is any d!fference between
a neutrino and an antineutrino. The neutral pion, as we shall see, is its own
antiparticle; so too is the photon. On the other hand, the antineutron is definitely
not the same as a neutron. So we're len in a bit of a quandary: is the neutrino
the same as the antineutrino, and if nol. what property distinguishes them?
In the late 1950s, Davis and Harmer put this question to an experimental test
[12J. From the positive results of Cowan and Reines, we know that the crosse<!
reaction
(1.12)
must also occur, and at about the same rate. Davis looke<! forthe analogous reaction
using antineutrinos:
(1.13)
• That's a comfortins ruliz.>tion wh�n you l�am thai hundrNs of billions of MUlrinos ptr =­
ond pass throujJh t:Y�ry squue inch ofyour body. nishl and day, cominS from the sun (they hit
you from below, at niSht, haYinS pas.=i right through the urth).
127
28 1 I HistQrir;o! !nlroduction
10 Ih. Elem.ntQry Parti"",
He found that this reaction does not occur. and concluded that the neutrino and
antineutrino are distinct particles.'
Davis's result was not unexpected. In fact, back in 1953 Konopinski and
Mahmoud [13J had introduced a beautifully simple rule for determining which
reactions - such as Equation 1.12 - will work, and which - like Equation 1.13 will not. In effect,t they assigned a !tpton number L "" + 1 to the electron, the muon,
and the neutrino, and L "" - 1 to the positron, the positive muon, and the antineu­
trino (all other particles are given a lepton number of zero). They then proposed
the law of canst/wlion of !tplon number (analogous to the law of conservation of
charge): in any physical process, the sum ofthe lepton numbers before must equal
the sum of the lepton numbers after. Thus the Cowan-Reines reaction (1.11) is
allowed (L = - 1 before and after), but the Davis reaction (1.13) is forbidden (on
the left L = -1, on the right L "" +1). It was in anticipation of this rule that I called
the beta decay particle (Equation 1.8) an antineutrino; likewise, the charged pion
decays (Equation 1.9) should really be written
(1.l4)
and the muon decays (Equation 1.10) are actually
J.L- _ e- + v + lI
J.L+ _ e+ + v + lI
(1.15)
You might be wondering what property distinguishes the neutrino from the
antineutrino. The cleanest answer is: !tpkm numbu - iI's + 1 for the neutrino and
- 1 for the antineutrino. These numbers are experimentally determinable, just as
electric charge is, by watching how the particle in question interacts with others. (As
we shall see, they also differ in their hdicity: the neutrino is 'left·handed' whereas
the antineutrino is 'right·handed'. But this is a technical matter best saved for later.)
There soon followed another curious twist to the neutrino story. Experimentally,
the decay of a muon into an electron plus a photon is never observed:
(1.16)
and yet this process is consistent with conservation of charge and conservation of
the lepton number. Now, a famous rule of thumb in particle physics (generally
• Actu.aUy, this condusion is nO! as fir�proof
as it once seemed. It could be the spin state
of this book. [ shall assume we ar� dealing
with Dirac neutrinos. but wr'll return to the
question in Ch�pter 11.
of the ii, r�th"r than th� fact Wt it is distinct
t Konopinski �nd Mahmoud 113) did not use
from v. WI forbids relction 1.13. Today. in
this terminology, �nd they got the muon as·
fact, there are two vi<lble rnodrb: Di= lie"'
signments wrong. But never mind. th� essen·
lri...n, which are distinct from their antiparti·
tial ide� was there.
cles. and Majora.... I"tt"'rinos. for which and
ii are two state. oftllt.l<l"'" particlt. For most
v
1.5 N�"trino5 (1930-1%2)
attributed to Richard Feynman) dedares that whatever is not expresslyforbidden is
mandatory. The absence of/l. -Jo e + y suggests a law ofconservation of 'mu-ness',
but then how are we to explain the observed decays /.I. -Jo e + v + v? The answer
occurredto a numberofpeople in thelate 1950s and early 19605 [14J: suppose there
are two different kinds of neutrino - one associated with the electron (v.) and one
with the muon (v,,). Ifwe assign a muon number L" = +1 to /.1.- and v". and
L" = -1 to /.1.+ and vi<' and at the same time an elecfron number -4 +1 to e- and
v. and r.. = -1 to e+ and v., and refine the conservation of lepton number into
two separate laws - conservation of electron number and conservation of muon
number - we can then account for aU allowed and forbidden processes. Neutron
beta decay becomes
=
.
(1.17)
the pion decays are
(1.18)
and the muon decays take the form
(1.19)
I said earlier thatwhen pion decay was first analyzed it was 'natural' and 'economi·
cal' to assume that the outgoing neutral particle was the same as in beta decay, and
that's quite troe: it was natural and it was economical, but it was wrong.
The first experimental test of the two·neutrino hypothesis (and the separate con­
servation ofelectron and muon number) was conducted at Brookhaven in 1962 [15J.
Using about 10l• antineutrinos from rr- decay, Lederman, Schwartz, Steinberger,
and their collaborators identified 29 instances of the expected reaction
(1.20)
and no cases of the forbidden process
(UI)
With only one kind of neutrino, the second reaction would be just as common
as the first. (Incidentally, this experiment presented truly monumental shielding
problems. Steel from a dismantled battleship was stacked up 44-feet thick, to make
sure that nothing except neutrinos got through to the target.)
I mentioned earlier that neutrinos are extremely light - in fact, until fairly
recently it was widely assumed (for no particularly good reason) that they are
1 29
"I I
HistoriUll lmroduclion 10 the £Iemenlory PortjC/",
Table 1.1 The lepton filmily, 1%2�1976
Lepton
Electron
Muon
number
number
number
Leptons
,-
".
0
•
""
Antileptons
,<
'.
.<
'"
0
0
0
-,
-,
,
-,
-
-,
-,
0
0
0
0
-
,
-,
massles.s This simplifies a lot of calculations, but we now know that it is not strictly
true: neutrinos haw: mass, though we do not yet know what those masses are,
except to reiterate that they are very small, even when compared to the electron's.
What is more, over long distances neutrinos ofone type can convert into neutrinos
again. in a phenomenon known as lUulnllO osciUation. But this story belongs much
ofanother type (for example, electron neutrinos into muon neutrinos) - and back
later, and deserves a detailed treatment, so I'U save it for Chapter 11.
By 1%2, then, the lepton family had grown to eight: the electron, the muon, their
respective neutrinos, and the corresponding antiparticles (Table 1.1). The leptons
are characterized by the fact that they do not participate in strong interactions. For
the next 14 years things were pretty quiet, as far as the leptons go, so this is a good
place to pause and catch up on the strongly interacting particles - the mesons and
baryons, known collectively as the Wrons.
1.'
Strange Particles (1947-1960)
For a brief period in 1947, it was possible to believe that the major problems of
elementary particle physics were solved. After a lengthy detour in pursuit of the
muon, Yukawa's meson (the 11") had finally been apprehended. Dirac's positron
had been found, and Pauli's neutrino, although still at large (and, as we have
seen, still capable ofmaking mischief), was widely accepted. The role ofthe muon
was something of a puzzle (,Who ordered t/ult?' Rabi asked) - it seemed quite
unnecessary in the overall scheme of things. On the whole, however, it looke<! in
1947 as though the job of elementary particle physics was essentially done.
But this comfortable state did not last long [16]. In December of that year,
Rochester and Butler [17] published the cloud chamber photograph shown in
Figure 1.8 Cosmic ray particles enter from the upper left and strike a lead plate,
�
U. SIron8t Parliclts (1947-1960)
�iCray
l�ide
,=
01 lead
Debri,
Fig.
1.7
The first strange putide. Cosmic rays strike i leid
�ate, producing a 1(0, wh ich subsequently decays into a piir
of charged pions. {Photo cOtlrtesy of Prof. Roches�r, G. O.
(CI 1947). Nature, 160. 855. Copyright Macmillin Journals
limited.)
producing a neutral particle, whose presence is revealed when it deays into two
charged secondaries, forming the upside·down 'Y' n
i the lower right. Detailed analy·
sis indicated that these charged particles are in fact a ]f+ and a]f- . Here, then, was a
newneutral particle withal least twice the mass ofthe pion: we call itthe JCl ('bon'):
(U2)
In 1949 Brown and her collaborators published the photograph reproduced in
Figure 1.8, showing the decay of a charged hon:
(1.23)
(The JCl was first known as the 01 and later as the 9°; the K+ was originally called
the r + . Their identification as neutral and charged versions of the same basic
particle was not completely settled until 1956 - but that's another story, to which
we shall return in Chapter 4.) The hons behave in some respects like heavy pions,
so the meson family was extended to include them. In due course, many more
mesons were discovered - the "fl, the .p, the (01, the p's, and so on.
Meanwhile, in 1950 another neutral 'Y' particle was found by Anderson's group
at Cal Tech. The photographs were similar to Rochester's (Figure 1.7), but this time
the products were a p+ and a ]f- . Evidently, this particle is substantially heavier
131
321 1 Hirt<>rictll lniroduefion
fO 1m Elementary Particles
.,,:
. /
"
.
. ,,.
. ,...;
. ,"..,
.
"./ :
debris
"
,.
,
,.'
,
\.
.'
,
,
'/
Fig. 1.8 WI-. entering from �bove. de<:�ys �t AK+ .... ".+
+ ".+ + "' - . (The ".- subsequently ClIuses a nudeir dis­
integr"tion it B.) (Source: Powell. C. F. Fowler. p, H. ind
Perkins, D. H. (1959) The Study of flemenlory Porticles by the
Photogrophi( Melhod. Pergamon. New York. First published
in Notu",. Hi3. 82 (1949).)
than the proton; we call it the A:
A _ p+ + :rr-
(U4)
The lambda belongs with the proton and the neutron in the baryon family. To
appredate this, we must go back for a moment to 1938. The question had arisen,
'Why is the proton stable?' Why. for example, doesn't it decay into a positron and a
photon:
(US)
1.6 SI","ge Porlid� (1947_J 960)
Needless to say, it would be unpleasant for us if this reaction were common (all
atoms would disintegrate), and yet it does notviolate any law known in 1938. (It does
violate conservation of Itpton number, but that law was not re.::ognized, remember,
until 1953.) Stiickelberg [18J proposed to account for the stability of the proton by
asserting a law of conservation of baryon number: assign to all baryons (which in
1938 meant the proton and the neutron) a 'baryon number' A
+1, and to the
antibaryons @andil) A= -I; thenthetotalbaryon nwnber is conserve<!in any physical process. Thus, neutron beta decay (n -jo p+ + �- + ii.) is allowed (A = 1 before
and after), and so too is the reaction n
i which the antiproton was first observed:
=
(1.26)
(A = 2 on both sides). But the proton, as the lightest baryon, has nowhere to go;
conservation ofbaryon number guarantees its absolute stability.' If we are to retain
the conservation ofbaryon number in the light ofreaction (1.24), the lambda must
be assigned to the baryon family. Over the nextfewyears, many more heavy baryons
were discovered - the :E's, the 3's, the .6.'s, and so on. By the way, unlike leptons
and baryons, there is no conservation of mesons. In pion de.::ay (n- -jo p. - + v).') a
meson disappears, and in lambda decay (A _ p+ + 11"-) a meson is created.
It is some measure of the surprise with which these new heavy baryons and
mesons were greeted that they came to be known collectively as 'strange' particles.
In 1952, the first of the modern particle accelerators (the Brookhaven Cosmolron)
began operating, and soon it was possible to produce strange particles in the
laboratory (before this the only source had been cosmic rays) . . . and with this
the rate of proliferation increased. Willis Lamb began his Nobel Prize acceptance
speech in 1955 with the following words [19J:
Wlu:n the Nobd Prize;s wu�flr5t awarded in 1901, physicists
kn�w sonuthing ofjusI two objects which ar� nOlll caUtd "elt·
mentary particlts": the dectron and /he proton. A deluge of olh­
u "tltnuntary" particlts appeared after 1930; neutron, neu·
trino, p. meson (sic}, n meson, hmviu mesons, and various hy­
perons. 1 have hmrd it said that "!he finder ofa ntlll eltnun·
tary paniclt used to be rewarded by a Nohtl Prize, but such a
disC<lvtry n(}\II ought to be punished by a $10,000fine".
Not only were the new particles unexpected; there is a more technical sense
in which they seemed 'strange': they are produad copiously (on a time s<:ale of
about 10-23 seconds), but they decay relatively slowly (typically about 10-10 sec·
onds). This suggeste<! to Pais and others [20J that the mechanism involved in
• 'Gr�nd unified theories' (GUTs) aUow fur a
minute violation of baryon num�r co,...r.., ·
vation. and in these theories the proton is
1\01 ab$olutdy stable (see S�tions 2.6 and
12.2). As of 2007, no proton d��y hu been
observed. and its lifetime is known to exceed
101' ye�I1I-whidt is pretty stable, wh= you
consider that the age of the unh'er5e is about
1010 �ars.
1 33
3<4 1
' Hisloricol lntroduction 10 th� Elementary PQrtid�
their production is entirely different from that which governs their disintegration.
In modern language, the strange particles are prodwctd by the strong force (the
same one that holds the nucleus together), but they dteay by the wtak force (the
one that accounts for beta decay and all other neutrino processes). The details of
Pais's scheme required that the strange particles be produced in pairs (so.called
associatro production). The experimental evidence for this was far from clear at
that time, but in 1953 Cell-Mann [211 and Nishiiima [22] found a beautifully
simple and, as it developed, stunningly successful way to implement and improve
Pais's idea. They assigned to each particle a new property (Cell. Mann called it
'strangeness') that (like charge, lepton number, and baryon nwnber) is conserved
in any strong interaction, but (unlike those others) is not conserved in a weak
interaction. In a pion-proton collision, for example, we might produce two strange
particles:
]f- +P+ _ K+ + E­
_ Ko + tO
_ JCl + A
(1.27)
=
=
Here, the K's carry strangeness S
+1, the t's and the A have S -1, and
the 'ordinary' particles - ]f , p, and n - have S = O. But we never produce just om:
strange particle:
]f- + p+ f. ]f+ + E­
fo ]f° + A
fo JCl + n
(1.28)
On the other hand, when these particles dteay, strangeness is not conserved:
A
_ P+ +]f ­
E + _ p+ + :n: o
_ 1I + :n: +
(1.29)
these are wtak processes, which do not respe<:t conservation ofstrangeness.
There is somearbitrariness in theassignment ofstrangeness numbers, obviously.
We could just as well have given S = + 1 to the t's and the A, and S = -1 to K+
and [(0; in fact, in retrospe<:t it would have been a little nicer that way. (In exactly
the same sense, Benjamin Franklin's original convention for plus and minus
charge was perfectly arbitrary at the time, and unfortunate in retrospe<:t, since
it made the cwrent<arrying particle - the electron - negative.) The significant
point is that there exists a consistent assignment of strangeness numbers to
all the hadrons (baryons and mesons) that accounts for the observed strong
processes and 'explains' why the others do not occur. (The leptons and the
1.7 Tlu f;g"tfold WQV (I96I-I964}
photon don't experience strong forces at all, so strangeness does not apply to
them.)
The garden that seemed so tidy in 1947 had grown into a jungle by 1960, and
hadron physics could onlybedescribedas chaos. Theplethora ofstrongly interacting
particles was divided into two great families - the baryons and the mesons - and
the members of each family were distinguished by charge, strangeness, and mass;
but beyond that there was no rhyme or reason to it all. This predicament reminded
many physicists of the situation in chemistry a century earlier, n
i the days before
the periodic table, when scores of elements had been identified, but there was no
underlying order or system. In 1960, the elementary particles awaited their own
'periodic table·.
1.7
The Eightfold Way (1961 -1%4)
The Mendeleev of elementary particle physics was Murray Gell·Mann, who intro­
duced the so-called Eightfold Way in 1961 [23J. (Essentially the same scheme was
proposed independently by Ne'eman.) The Eightfold Way arranged the baryons and
mesons into weird geometrical patterns, according to their charge and strangeness.
The eightlightestbaryons fit intoa hexagonal array, withtwoparticles atthe center:'
"
p
___
S : O - - - . . - - - - __�
,"
•
•
A
S z -I - _ _ ...... �-
s - - ' -.
,
- -- - - - �":---____
,
,
0 ,, -1
The baryon octet
,.
\
,
,
,
,
,
,
a-o
,
,
,
,
0 "' + '
This group is known as the baryon octet. Notice that particles oflike charge lie along
the downward-sloping diagonal lines: Q = +1 (in units of the proton charge) for
the proton and the 1:+; Q = 0 for the neutron, the A, the 1;0, and the ;:;0; Q -1
for the r:- and the ;:;-. Horizontal lines associate particles oflike strangtntss: 5 = 0
for the proton and neutron, S "", -1 for the middle line, and 5 = -2 for the two ;:;'s.
The eight lightest mesons fill a similar hexagonal pattern, forming the (pseudo­
scalar) nu:son octet:
=
• Th� �bti� pbc�=nt of the p.arlicles in the cen�r is arbitrary. but in this book I shall �lways
put the neutr�l member of the triplet (here the r:o) above th� singlet (her� the A).
1 15
361 1
H;slori«Jl lntrod",;1;on 10 1h� fl.mon1ory POr/je/os
K"
,
,
"
•
•
,
5 - 0 - - - - - "'
5 - - 1 - - - - - - - - - \,
,
,
"
,
K'
,
0 .. - 1
,
,
,
The meson octe1
,
,
,
,
,
,
,
,
,
a- ,
0-0
Once again, diagonal lines determine charge and horizontal lines determine
= 0, and the
= -1. (This discrepancy is again a historical accident; Gell-Mann
could just as well have assigned S = I to the proton and neutron, S
0 to the
strangeness, but this time the top line has S = 1, the middle line S
bottom line S
=
1:'s and the A, and S "" -1 to the 3's. In 1953 he had no reason to prefer that
choice, and it seemed most natural to give the familiar particles - proton, neutron,
and pion - a strangeness of zero. After 1961, a new tenn - hypercharxe - was
introduce<!, which was equal to S for the mesons and to S + 1 for the baryons. But
tater developments revealed that strangeness was the better quantity after all, and
the word 'hypercharge' has now been taken over for a quite different purpose.)
Hexagons were not the only figures allowed by the Eightfold Way; there was
also, for example, a triangular array, incorporating 10 heavier baryons - the baryon
dtcuplet:*
lI + +
lI '
alI O
5 - 0 - - - - -- <;��_"
--"
-
,
r; "
5 - - 1 - - - - .....
\
,
,
\
,
0 .. - 1
,
,
,
,
,
,
\
'
0-2
,
0-'
0 "0
• In this book, for simplicity, ! �dhere to the old.f�shioned no�!ion in which the decuplel \»<1i·
c� �� design�ted E* �nd S*: modem US<.ge drops the s�r �nd pUIS the mUS in parentheses:
E(138S) �nd 8(1530).
1.8 The Qua,* Model (1964)
Now. as Gell-Mann was fitting these particles into the decuplet. an absolutely
lovely thing happened. Nine of the particles were known experimentally. but
at that time the tenth particle (the one at the very bottom. with a charge of
- 1 and strangeness 3) was missing; no particle with these properties had
ever been detected in the laboratory [24J. Gell-Mann boldly predicted that such
a particle would be found. and told the experimentalists exactly how to pm·
duce it. Moreover. he calculated its mass (as you can for yowself. in Problem
1.6) and its lifetime (Problem 1.8) - and sure enough. in 1964 the famous
omega-minus particle was discovered [251. precisely as Gell-Mann had predicted
(see Figure 1.9).'
Since the discovery of the omega-minus (n-). no one has seriously doubted
that the Eightfold Way s
i correct. Over the next 10 years. every new hadron
found a place in one of the Eightfold Way supentlultipkts. Some of these are
shown in Figure 1.10t . In addition to the baryon octet. decuplet, and so on.
there exist of course an antlbaryon octet. decuplet. etc. with opposite charge
and opposite strangeness. However. in the case of the mesons. the antiparticles
lie in the same surxntluhip/et as the corresponding partides. in the diametri­
cally opposite positions. Thus the antiparticle of the pi-plus s
i the pi-minus. the
anti·K·minus is the K-plus. and so on (the pi·zero and the eta are their own
antiparticles).
Classification s
i the first stage in the development of any science. The Eightfold
Way did more than merely classify the hadrons. but its real importance lies in the
organizational structure it provided. I think it's fair to say that the Eightfold Way
initiated the modern era in particle physics.
-
1.8
The Quark Model (1964)
But the very success of the Eightfold Way begs the question: why do the hadrons
fit into these bizarre patterns? The periodic table had to wait many years for
quantum mechanics and the Pauli exclusion principle to provide its explanation.
An understanding of the Eightfold Way. however. came already in 1%4. when
Gell-Mann and Zweig independently proposed that all hadrons are in factcomposed
of even more elementary constituents. which Gell-Mann called quarks 126]. The
• A simil�r thing happened in the case of the periodic table. There were three famous 'holes'
(missing dements) on Mendd�s ch<irt. and he predicted that new elements would � discov·
ered to fill in the gaps. Like Gell·Mann. he confidently described their properties. and within 20
Y"ars all three - gallium. scandium. and germanium - wer� found.
t To � sure. there were occasional false alarms particles that did not seem 10 fi! Gell.Mann·s
sch�m� - but they always turned out to � experimental errors. Elementary paredes haye a w�y
of appearing and then dw.ppeanng. Of the 26 mesons listed on a standard table in 1%3. 19
were la�r found to � spurious!
_
131
38
'.
I 1 HilUJrical lntroduClioro UJ the flemet1tary
Parlides
-
-
t,
--
-
-
,
•
'" . G, \::.
\::;, 0
r.;\..:Y &'.
1.8 Th� QIlOrk Model (1964)
@
KO
K'
"
K
r;;;\ 1(0'
,,0\,.
\:..Y
1(-
••
KO
Fig. 1.10 Some meson nonets, labeled in
s�ctroscopic notation (see Chapter 5).
There are now at least 15 established nonels
(though in some cases not all members
hve been diKovered). For the baryons there
I. f.
a.
.
.
a,
1(,',
1(,'.
1(,-.
/(,"..
are Ihr� complete octets (with spins 1/2,
3/2, and 5/2) and 10 others partly �lled;
Ihere is only one complete decuplel, but 6
mOre are partly filled, and there art thr�
known singlets.
'57
quarks come in three types (or flavors') forming a triangular 'Eightfold.Way'
'
pattern:
- . , - ----- -
.
"
d
\
.-- 1 ------ __
,
,
,
,
,
,
..;
Q. '
\,
.,
The quarks
\0 - --}
The II (for 'up') quark carries a charge of
� and a strangeness ohero; the d ('down')
-.' - - �
- --
quark carries a charge of -
i n
i ally 'sideways', but now more
� and S = 0; the s (org
-!
commonly 'strange') quark carries a charge of
and S = -1. To each quark (q)
there corresponds an antiquark @, with the opposite charge and strangeness:
$ - 0 ------ __
\
\
Ii
if
,
,
,
The antiquarl<.s
,
,
,
,
,
a - -t
,
,
,
,
,
,
o- t
1 39
40
I'
Histori«Ji /ntrodllclion 10 the EI�mMI(l'l' P(lrtides
And there are two compositioll rules:
1. Every baryon is composed of three quarks (and every
alllibaryon is compose<! of three allliquarks).
2. Every meson is compose<! of a quark and an antiquark.
With this, it is a matter ofelementary arithmetic to constructthe baryon decuplet
and the meson octet. All we need to do s
i list the combinations ofthree quarks (or
quark-antiquark pairs) and add up their charge and strangeness:
The baryon decuplet
'"
'"
"""
,dd
ddd
''''
Q
s
Baryon
2
1
0
-1
1
0
0
0
0
0
-1
6"
dd>
-1
-1
-1
=
0
-1
-2
-2
-1
-3
"",
d»
m
6+
6"
6-
E '+
E-
E'-
8-
S·W
Notice that there are 10 combinations of three quarks. Three u's, for instance,
�
at Q = each, yield a total charge of +2 and a strangeness of zero. This is the
t:. ++ particle. Continuing down the table, we find all the members of the decuplet
ending with the Q-, which is evidently made ofthree s quarks.
A similar enumeration of the quark-antiquark combinations yields the meson
table:
qij
od
'"
dd
'"
'"
dl
,;;
�
'"
The meson nonet
Q
s
0
1
-1
0
1
0
-1
0
0
0
0
0
0
Meson
'
rr
rr+
rr
,
K'
K+
-1
-1
0
K"
K??
1.8 The Quark Model (1964)
But wait! There are nine combinations here, and only eight particles in the meson
octet. The quark model requires that there be a third meson (in addition to the H I)
and the '7) with Q = 0 and S = O. As it turns out, just such a particle had already
been found experimentally - the '7'. In the Eightfold Way, the 1]' had been classified
as a singlet, all by itself. According to the quark model, it properly belongs with the
other eight mesons to form the meson nonel. (Actually, since uli, dd, and ss all have Q
= 0 and S = 0, it is not possible to say, on the basis ofanything we have done so far,
which is the HO, which the 1], andwhich the 1/. Butnever mind, the point is that there
are three mesons with Q = S = 0.) By the way, the antimesons automatically fall in
the same supennultiplet as the mesons: ud is the antiparticle of dU, and vice versa.
You may have noticed that I avoided talking about the baryon octet - and it is
far from obvious how we are going to get eight baryons by putting together three
quarks. In truth, the procedure is perfectly straightforward, but it does call for some
facility n
i handling spins, and I would rather save the details for Chapter 5. For now,
I'll just tantalize you with the mysterious observation that if you take the decuplet
and knock offthe three corners (where the quarks are identical - uuu, ddd, and sss)
and double the center (where aU three are different - uds), you obtain precisely the
eight states in the baryon octet. $0 the same set of quarks can account for the octet;
it's just that some combinations do not appear at all, and one appears twice.
Indeed, aU the Eightfold Way supetmultiplets emerge naturally in quark model.
Of course, the same combination of quarks can go to make a number of different
particles: the delta·plus and the proton are both composed of two u's and a d; the
pi-plus and the rho.plus are both ud. and so on. Just as the hydrogen atom (electron
plus proton) has many different energy levels, a given collection of quarks can
bind together in many different ways. But whereas the various energy levels in
the electron/proton system are relatively close together (the spacings are typically
several electron volts, n
i an atom whose rest energy is nearly 109 eV), so that we
naturally think of them all as 'hydrogen', the energy spacings for different states
of a bound quark system are very large, and we normally regard them as distinct
particles. Thus we can, in principle, construct an infinite number ofhadrons out of
only three quarks. Notice, however, that som� things are absolutely excluded in the
quark model: for example, a baryon with S = 1 or Q = -2; no combination of the
three quarks can produce these numbers (though they do occur for antibaryons).
Nor can there be a meson with a charge of+2 (like the L'r. ++ baryon) or a strangeness
of -3 (like the frIo For a long time. there were major experimental searches for
these so-called 'exotic' particles; their discovery would be devastating for the quark
model, but none has ever been found (see Problem 1.11).
The quark model does, however, suffer from one profound embarrassment:
in spite of the most diligent search, no one has ever seen an individual quark.
Now, if a proton is really made out of three quarks, you'd think that if you hit
one hard enough, the quarks ought to come popping out. Nor would they be
hard to recognize, carrying as they do the unmistakable fingerprint of fractional
charge - an ordinary Millikan oil drop experiment would clinch the identification.
Moreover, at least one ofthe quarks should be absolutely stable; what could it decay
1 .1
42 1 ' HistoricollntrodUclion to th� £1�m�"lory Portides
\
•
..
,
!
z
1
•
\
'.
•
\
�
\
•
..
'.
'.
'.
, .
AIQm has
substructure
\ .•
\ '.
.
\
\
\
PrOtQ" has
substructure
\
.
\ ....
.
\
•
\ ++
1.1
Seallering angle
Fig. 1.11 (�) In Rutherford suttering, the
number of particles defle<:ted through large
angles indicates that the �tom has internal
structure (i nucleus). (b) In deep inelastic
scatttring. the number of particles defle<:ted
through large angles indicates that the pro·
ton has internal structure (quarks). The
dashed lines show what )'QU would expect
fbi
if the positive ch�rge were uniformly dis·
tributed over Ihe volume Qf (a) lhe atom,
(b) the proton. (Soma: Halzen, F. and Mar·
lin. A. O. (1984) Quarks ond Lep1Ol15. John
Wiley & Sons, N_ York, p. 17. Copyright e
John Wiley & Sons. Inc. Reprinled by permis·
sion.)
into, since there is no lighter particle with fractional charge? So quarks ought to be
easyto produce, easy to identify, andwsy to store, andyet, no one has ever foundone.
The failure of experiments to produce isolated quarks occasioned widespread
skepticism about the quark model in the late 1960s and early 1970s. Those who
dung 10 the model tried to conceal their disappointment by introducing the notion
of quark con,fintmmt: perhaps, for reasons not yet understood, quarks are absolutely
wnftntd within baryons and mesons, so that no matter how hard you try, you
cannot get them out. Of course, this doesn't explain anything, it just gives a
name to our frustration. But it does pose sharply a critical theoretical question
that is still not completely answered: what is the mechanism responsible for quark
confinement? [27]
Even if all quarks are stuck inside hadrons, this does not mean they are
inaccessible to experimental study. One can explore the interior of a proton in
much the same way as Rutherford probed the inside ofan atom - by firing things
into it. Such experiments were carried out in the late 1960s using high-energy
electrons at the Stanford Linear Accelerator Center (SLAC). They were repealed in
the early 1970s using neutrino beams at CERN, and later still using protons. The
results ofthese so·called 'deep inelastic scattering' experiments [28] were strikingly
reminiscent of Rutherford's (Figure 1.11): most ofthe inddent particles pass right
through, whereas a small number bounce back sharply. This means that the charge
ofthe proton is concentrated in smaU lumps, just as Rutherford's results indicated
that the positive charge in an atom s
i concentrated at the nucleus [29]. However,
/.8 Th� QUQm Model (1964)
in the case ofthe proton the evidence suggests three lumps, instead of on�. This is
strong support for the quark model, obviously, but still not conclusive,
Finally, there was a theoretical obi«:tion to the quark model: it appears to vi­
olate the Pauli exclusion principle. In Pauli's original formulation, the exclusion
principle states that no two electrons can occupy the same state. However, it was
later realized that the same rule applies to all particles of half·integer spin (the
proof of this is one of the most important achievements ofquantum field theory).
In particular, the exclusion principle should apply to quarks, which, as we shall
see, must carry spin i Now the 1:1.++, for instance, is supposed to consist of three
identical u quarks in the same state; it (and also the 1:1.- and the n-) appear to be
inconsistent with the Pauli principle, In 1964, O. W. Greenberg proposed a way
out of this dilemma ]30]. He suggested that quarks not only come in three jlavo�
(u, d, and 5) but each of these also comes in three colo� ('re<l', 'green', and 'blue',
say). To make a baryon, we simply take one quark ofeach color; then the three u's
in 1:1.++ are no longer identical (one's red, one's green, and one's blue). Since the
exclusion principle only applies to idmtical particles, the problem evaporates.
The color hypothesis sounds like sleight of hand, and many people initially
considered it the last gasp of the quark model. As it turned out, the introduction of
color was extraordinarily fruitful [31]. I need hardly say that the term 'color' here
has absolutely no connection with the ordinary meaning of the word. Redness,
blueness, and greenness are Simply labels used to denote three new properties that,
in addition to charge and strangeness, the quarks possess. A red quark carries
one unit of redness, zero blueness, and zero greenness; its antiparticle carries
minus one unit of redness, and so on. We could just as well call these quantities
X·ness, Y-ness, and Z'ness, for instance, However, the color terminology has one
especially nice feature: it suggests a delightfully simple characterization of the
particular quark combinations that are found in nature.
All naturally occurring particles are colorless.
By 'colorless' I mean that eillKr the total amount of each color is zero or all
three colors are present in equal amounts. (!be latter case mimics the optical
fact that light beams of three primary colors combine to make white.) This clever
rule 'explains' (if that's the word for it) why you can't make a particle out of two
quarks, or four quarks, and for that matter why individual quarks do not occur in
nature. The only colorless combinations you can make are qij (the mesons), qqq
(the baryons), and ijijij (the antibaryons).'
• Of cou�. you Gin p.ack.a� tog�th�r combi.
""Iio.... of th�� - th� d�ut�ron. for a:l.mpl�,
is a six qu.ark sute (three ,,'s and three d·s).
In 2003, th� was a Hurry of excitement
OYer the app.arent observation of four·quark
'm�soll5' (acrnally, 9qij1j} and I"'ntaquark
'bolT)'<lns' (qqqqij). The latter nOW appe<lr to
have been statistic<ll artifacts 1321, but in at
l�aS! one m�son ca� (th� so-callffi X(3872)
discovered <It KEK in lapan). th� four-qUllrk
int�rpr�tation �ms to t... holding up, though
it is still not c1�ar wh�ther it is best thought
of as a DD> 'mol�ul�' or as a m�n in its
own right 1331_
143
«1 1
His!oricol lnlroduction to the Elemenlary Particles
1.'
The November Revolution and Its Aftermath (1974-1983 and 1995)
The de<ade from 1964 to 1974 was a barren time for elementary particle physics.
The quark model, which had seemed so promising at the beginning, was in
an uncomfortable state of limbo by the end. It had some striking successes: it
neatly explained the Eightfold Way, and correctly predicted the lumpy structure
of the proton. But it had two conspicuous defects: the experimental absence
of free quarks and inconsistency with the Pauli principle. Those who liked the
model papered over these failures with what seemed at the time to be rather
transparent rationalizations: the idea of quark confinement and the color hy·
pothesis. But I think it is safe to say that by 1974 most elementary particle
physicists felt queasy, at best. about the quark model. The lumps inside the pro·
ton were called partons. and it was unfashionable to identitY them explicitly with
quarks.
Curiously enough, what rescued the quark model was not the discovery of
free quarks, or an explanation of quark confinement. or confirmation of the
unexpected: the discovery ofthe psi meson. The t/I was first observed at Brookhaven
color hypothesis. but something entirely different and (almost) [34] completely
by a group under C. C. Ting, in the summer of 1974. But Ting wanted to check
his results before announcing them publicly. and the discovery remained an
astonishingly well.kept set:ret until the weekend of November 10- 11. when the
The two teams then published simultaneously [35]. Ting naming the particle l,
new particle was discovered independently by Burton Richter's group at SLAC.
and Richter calling it t/I. The IN was an elet:trically neutral. extremely heavy
meson - more than three times the weight of a proton (the original notion that
mesons are 'middle·weight' and baryons ·heavy·weight' had long since gone by the
for the t/I lasted fully
boards). But what made this particle so unusual was its extraordinarily long lifetime.
seconds before disintegrating. Now, 10-l() set:onds may
lO-Ul
not impress you as a particularly long time. but you must understand that the
typical lifetimes for hadrons in this mass range are on the order of 10-2) seconds.
So the t/I has a lifetime about a 1000 times longer than any comparable particle. It's
as though someone came upon an isolated vi
l lage in Peru or the Caucasus where
would be a sign offundamentally new biology at work. And so it was with the 1/t: its
people live to be 70 000 years old. That wouldn't just be some actuarial anomaly. it
good reason, the events pret:ipitated by the discovery ofthe t/I came to be known as
long lifetime, to those who understood. spoke of fundamentally new physics. For
In the months that followed, the true nature of the t/I meson was the subjet:t of
the November Rtvolution [36].
t/I s
i a bound state of a new (fourth) quark, the c (for charm) and its antiquark,
t/I = (cC). Actually, the idea of a fourth flavor, and even the whimsical name, had
ively
l
debate, but the explanation that won was provided by the quark model: the
been introduced many years earlier by Bjorken and Glashow (37). There was an
1.9 The NCMm�r Re"olu!jon ondII,Aftermath (1974-1983 and 1995)
intriguing parallel between the leptons and the quarks:
uptons : e, v" /-I., vI'
Quarks : d, U,S
If all mesons and baryons are made out ofquarks, these two families are left as the
tmly fundamental particles. But whyfour leptons and only three quarks? Wouldn·t
it be nicer if there were four of each? Later, Glashow, !liopoulos, and Maiani [38]
offered more compelling technical reasons for wanting a fourth quark, but the
simple idea ofa parallel between quarks and leptons is another of those far-fetched
spe<ulations that turned out to have more substance than their authors could have
imagined.
So when the 1/1 was discovered, the quark model was ready and waiting with an
explanation. Moreover, it was an explanation pregnant with implications. For if a
fourth quark exists, there should be all kinds ofnew baryons and mesons, carrying
various amounts ofcharm. Some ofthese are shown in Figure 1.12; you can work
out the possibilities for yourself (Problems 1.14 and 1.IS). Notice that the ", itself
;"
-re
(a)
-re
;-
D"
K"
,
,-
=
(b)
;"
-re
�
5°
fl� r;
..
-
+
s:c
,
dr
If
•
D"
{]�e�
...
,
'"'
01:'
••
fl-
,
,.
,�
5°
d-
1:"
Fig. 1.12 Su�,multiplets constructed using four·qu�rk fla­
vors: b�ryons (0 and b) and mesons (c �nd d). (Souru:
Review of Particle Phy,iQ.)
,"
(d)
p"
1:,
�
,
(e)
D;
145
<461 I HistoriUll lntroduaion to the EiemMUl"t' Particles
.
•
'.
,
•
.
•
.
•
'.
/
..
1.10 Inrrrmrdio/e Vector Bosom (19S3)
carries no nd charm, for ifthe c is assigned a charm of+ I, then c will have a charm
of-I; the charm ofthe '" is, ifyou will, 'hidden', Toconnrm the charm hypothesis,
it was important to produce a particle with 'naked' (or 'bare') charm [39J. The nrst
evidence for charmed baryons (At = udc and Ei+ = ULU) appeared already in
1975 (Figure 1.13) [40J, followed later by S, = usc and Q, = 5:;(, (In 2002 there were
(00 = cu and 0+ = cd) were discovered in 1976 (411, followed by the charmed
hints of the nrst doubly charmed baryon at Fermilab.) The nrst charmed mesons
strange meson (OJ = d) n
i 1977 [42J. With these discoveries, the interpretation of
the ", as c, was established beyond reasonable doubt. More important, the quark
model itself was put back on its feel.
i 1975 a new lepton was discovered
However, the story does not end there, for n
(43J, spoiling Glashow's symmetry. This new partide (the tau) has its own neutrino,
so we are up to six leptons, and only four quarks, But don't despair, because 2 years
later a new heavy meson (the upsilon) was discovered [44J, and quickly recognized
as the carrier of a nfth quark, b (for uauty, or bottom, depending on your taste):
""( = bb. Immediately the search began for hadrons exhibiting 'naked beauty', or
'bare bottom.' (I'm sorry. I didn't invent this terminology. In a way, its silliness is
A� = udb, was observed in the 1980's, and the
a reminder of how wary people were of taking the quark model seriously, in the
early days.) The first bottom baryon,
second (Et = uub) in 2006; in 2007 the first baryon with a quark from all three
generations was discovered (Sh" = dsb), The nrst bottom mesons (SO = bd and
B- = bU) were found in 1983 [45J. The Ef!{Tf! system has proven to be especially
rich, and so...::alled 'B factories' are now operating at SUC ('BaBar') and KEK
('Belle'). The Particle Physics Bookkt also lists d1 = sb and Bt = cb.
At this point, it didn't take a genius to predict that a sixth quark (t, for lruth,
of course, or top) would soon be found, restoring Glashow's symmetry with six
and frustratingly elusive (at 174 GeV{e2, it is over 40 times the weight of the
quarks and six leptons. But the top quark turned out to be extraordinarily heavy
T) were unsuccessful, both because the electron-positron colliders did not reach
bottom quark). Early searches for 'toponium' (a Ii meson analogous to the ", and
i simply too
high enough energy and because, as we now realize, the top quark s
short·lived to form bound states - apparently there ar, no top baryons and mesons.
The top quark's existence was not definitively established until 1995, when the
previous year [46]. (The basic reaction is u + u (or d + d) ...... t + i; the top and
Tevatron finally accumulated enough data to sustain strong indications from the
anti.top immediately decay, and it is by analyzing the decay products that one is
able to infer their fleeting appearance.) Until the LHC begins operation, Fermilab
will be the only accelerator n
i the world capable ofproducing top quarks.
1,10
Intermediate Vector Bosons (1983)
In his original theory ofbeta decay (1933), Fermi treated the process as a contact
n
i teraction, occurring at a single point, and therefore requiring no mediating
147
'" J 1
Hincrical l"lroduClion 10 rhe Elemenlary Particles
particle. As it happens, the weak force (which s
i responsible for beta decay) is of
extremely shortrange, so that Fermi's model was not far from the truth, and yields
excellent approximate results at low energies. However, it was widely recognized
that this approach was bound to fail at high energies and would eventuaUy have
to be supplanted with a theory in which the interaction is mediated by the
exchange of some particle. The mediator came to be known by the prosaic name
inlermediale veclor boson, The challenge for theorists was to predict the properties
of the intermediate vector boson, and for experimentalists, to produce one in the
laboratory. You may recall that Yukawa, faced with the analogous problem for the
strong force, was able to estimate the mass of the pion in terms of the range of
the force, which he took to be roughly the same as the size of a nucleus. But we
have no corresponding way to measure the range of the weak force; there are no
'weak bound states' whose size would n
i form us - the weak force is simply too
feeble to bind particles together. For many years, predictions of the intennediate
vector boson mass were little more than educated guesses (the 'education' coming
largely from the failure of experiments at progressively higher energies to detect
the particle). By 1%2, it was known that the mass had to be at least half the
proton mass; 10 years later the experimental lower limit had grown to 2.5 proton
masses.
But it was not unti
l the emergence of the electroweak theory of Glashow,
Weinberg, and Salam that a really firm prediction of the mass became pos·
sible. In this theory, there are in fact thru intermediate vector bosons, two
of them charged (W*) and one neutral (2). Their masses were calculated to
be [47J
Mw = 82 ± 2 GeV/C-,
Mz = 92 ± 2 GeV/c2
(predicted)
(1.30)
[n the late 1970s, CERN began construction of a proton-antiproton collider de·
signed specifically to produce these extremely heavy particles (bear in mind that
the mass of the proton is 0.94 GeVlc2, so we're talking about something nearly
100 times as heavy). In January 1983, the discovery ofthe W was reported by Carlo
Rubbia's group [48J, and 5 months later the same team announced discovery ofthe
Z [49]. Their measured masses are
Mw = 80.403 ± 0.029 GeV;C-,
Mz = 91.188 ± 0.002 GeV;c-
(measured)
(1.31)
These experiments represent an extraordinary technical triumph [50J, and they
were of fundamental importance in confirming a crucial aspect of the Standard
Model. to which the physics community was by that time heavily committed (and
for which a Nobel Prize had already been awarded). Unlike the strange particles
or the "" however, (but like the top quark a decade later) the intennediate vector
bosons were long awaited and universally expected, so the general reaction was a
sigh of relief, not shock or surprise.
1. J I The Slandard Model (1978-?)
1.11
The Standard Model (1978-?)
In the current view, then, all matter is made out of three kinds of elementary
particles: leptons, quarks, and mediators. There are six leptons, classified according
to their charge (Q), electron number (4), muon number (L,,), and tau number
(Lt). They faU naturally into three gwerations:
Lepton classification
I
,
First generation
'.
Second generation
�
I
I
0
I
L,
4
0
0
0
0
"
"
I
0
I
0
0
0
I
0
'.
I
0
0
I
0
0
0
I
,
Third generation
Q
There are also six antileptons, with all the signs reversed. The positron, for example,
carries a charge of +1 and an electron number - I . So there are really 12 leptons,
Similarly, there are six 'flavors' of quarks, classified by charge, strangeness (S),
all told.
charm (q, beauty (B), and truth (T). (For consistency, I suppose we should include
redundant, inasmuch as the only quark with 5 = C = B = T = 0 and Q =
'upness', U, and 'downness', D, although these terms are seldom used. They are
�, for
instance. is the up quark, so it is not necessary to specifY U = 1 and D = 0 as well.)
The quarks, too. faU into three generations:
Quark classification
First generation
Second generation
Third generation
q
d
"
,
,
b
,
Q
'13
213
11 3
213
'1 3
213
D
U
S
C
B
I
0
0
0
0
0
0
I
0
0
0
0
T
0
0
I
0
0
0
0
0
0
I
0
0
0
0
0
0
I
0
0
0
0
0
0
I
Again. all signs would be reversed on the table of antiquarks. Meanwhile, each
quark and antiquark comes n
i three colors, so there are 36 ofthem in all.
Finally, every interaction has its mediator - the photon for the electromagnetic
force, two W's and a Z for the weak force, the graviton (presumably) for gravity
. . . but what about the strong force? In Yukawa's original theory the mediator of
strong forces was the pion. but with the di scovery of heavy mesons this simple
and K's and </J's and all the rest ofthem. The quark model brought an even more
picture could not stand; protons and neutrons could now exchange p's and I)'S
radical revision: for if protons, neutrons, and mesons are complicated composite
I.'
50
I I His«>riUli I"troductio" !O lhe Elemental)' PQrfides
'-b
'.
< '
,
"
d
•
'.
,,
"
3
m
,
Fig. 1.14 Th� tilr� g�ntr�tion§ of quarks and leptons, in order of increasing mass.
structures, there is no reason to believe their interaction should be simple. To study
the strong force at the fundamental level. one should look, rather, at the interaction
between individual quarks. So the question becomes: what particle is exchanged
between two quarks, in a strong process? This mediator is called the gluon, and in
the Standard Model there are eight ofthem. As we shall see, the gluons themselves
carry color, and therefore (like the quarks) should not exist as isolated particles. We
can hope to detect gluons only within hadrons or in colorless combinations with
other gluons (ghuoolls). Nevertheless, there is substantial indirect experimental
evidence for the existence of gluons: the deep inelastic scattering experiments
showed that roughly half the momentum of a proton is carried by electrically
neutral constituents, presumably gluons; thejet structure characteristic of inelastic
scattering at high energies can be explained in terms ofthe disintegration ofquarks
and gluons in flight IS 1J and glueballs may conceivably have been observed [52].
This is all adding up to an embarrassingly large number of supposedly 'elemen·
tary' particles: 12 leptons, 36 quarks, 12 mediators (I won't count the graviton,
since gravity is not included in the Standard Model). And, as we shall see later, the
Glashow-weinberg-Salam theory calls for at least one Higgs particle, so we have
a minimum of 61 particles to contend with. Informed by our experience first with
atoms and later with hadrons, many people have suggested that some, at least, of
these 61 must be composites of more elementary subparticles (see Problem 1.18)
[53[. Such speculations lie beyond the Standard Model and outside the scope of
this book. Personally, [ do not think the large number of 'elementary' particles in
the Standard Model is by itselfalarming, for they are tightly interrelated. The eight
gluons, for example, are identical except for their colors, and the second and third
generations mimic the first (Figure 1.14).
Still, it does seem odd that there should be three generations of quarks and
leptons - after all, ordinal'}' matter is made of up and down quarks (in the form of
protons and neutrons) and electrons, all drawn from the first generation. Why are
there two 'extra' generations; who needs 'em? It's a peculiar question, presuming
a kind of purpose and efficiency on the part of the creator for which there is
1.11 Th. SflJndard Mmiel (1978-?)
lti tle evid en ce . . , but on e can ', help won d ering . Actual ly, ther e is a surpr s
i ing
ans wer: as we s hall s ee, the pr edo minan ce of mat ter o ver anti matter ad mits a
plausibleaccou n ting within the Standard Mod el , butonl y if there are (at least)
three gen era to
i n s.
Of cours e,this begs ther evers e qu est o
i n: whyar eth er e only three generations?
Indeed, could there be more ofth em, which have not yetbeen dis co vered (presum.
ably becau setheya retoo heavyto be mad e with existing ma chin es )? As r ecentlyas
1988 (54), there wer egood r easons to anti cipat ea fourthg en eration ,and p er haps
i a year that possibility was foreclosed by expe ri ments at
even a fifth. But wit hn
SLAC and CE RN (55J. The z!! is (as Sadda m would s a y) th e 'mother of allparti cl es' ,
in the s ens e that it can d eca y (with a pr ec is el y calculable probability) n
i to any
qua rk /a ntiquark o r lepton /an til epton pa ir (e- e+, U + ii, vp + "ii", et c.), pro vi
only that the particle's mass is less than halfthat ofthe z!! ( e
enough en erg yto mak ethe pair). So by measuringthel if etimeof the z!! youcan
a ctuall y countthen u mber of quarks and l eptons with mass l ess than 45 GeV/c2.
Th e mo r ether eare,thes ho rt er the lifetimeofthe z!!, justas the more fatal diseases
we are susceptibleto the shortero ur average lifespan becomes. The exp eri ments
show that the lifetime ofthe z!! is exa ctly what you would exp ect on thebasis of
the established threeg en er a tons
i
. Of cours e,the quarks (an d conceivably even the
charged lepton) ina p utat ive fourth generation might be tooheavyto a ffect the ZO
lifetime, but it is ha rdl y to be ima gined that thefou r th n eut rino would sudden ly
jumpto o ver 45 GeVlc1. Atan yrate, what the exp eri ments doun equ i vo call yshow
is thatthe number of light neutrinosis 2.99 ± 0.06.
+
Althoughthe Sta ndard Mod el hass urvived uns cathed for 30 years ,it is certain ly
not the end ofthestor y. Th er ea r e man y impo rtant s
i s ues that it s impl y does not
add ress - it d oes not , for exa mpl e, tell usho w to calculate the quarkand lepton
masses."
Quark and l epton mass es ( i
lepton
="
quark
"
'.
"
,
"
,
<2 x 10<0.2
<18
0,511
106
1777
"
d
,
,
b
,
mass
2
5
100
1200
4200
174000
In the Standard Model, these are simply empi ri cal nu mbers , taken from exp eri·
ment , but a mature theory. presuma bly, wo uld explain them, just as we can for
theato ms on th ep er o
i dic tabl e.t As weshall s ee,th e Standard Model also tak es as
emp irical inp ut threeangles and aphas e in the Koba yashi-Maskawa matrix, and
analogousnu mbe rs fo r theleptons,and the Weinbergan gled escribin g electroweak
• There is substantial uncertainty in th� light quark masses; I haw. rounded them off for th� sake
of darity.
t Note. how�r. that the quark/lepton mass formw.. is going to look very strang<:, since it hu to
rover a range of at least 11 powes of 10. from the electron neutrino to th� top quark.
1 51
521 1 Hislorico! I"troduction 10 Ih, E!ementD'l' Panides
i the Standard
mixing, and . . . all told, there are over 20 arbitrary parameters n
Model, and this is simply unacceptable in any 'final' theory [56[.
On the experimental side, there remains much to be learned about neutrino
oscillations (see Chapter 11), and CP violation (Chapter 12), but the most conspic­
uous missing link is the Higgs particle, which is n«essary in the Standard Model
to account for the masses of the W and Z (and perhaps all other particles as well).
Like the top quark. the predicted mass of the Higgs has increased with time, as
each new experiment failed to discover it. At this point. it is presumably beyond the
range of any existing accelerator, and, since the cancellation of the SSC, the LHC
is ow best hope for finding this elusive particle.
Meanwhile, there are a number of theoretical speculations (supported as yet by
no dir«t experimental evidence) that go beyond the Standard Model. There are the
Grand Unified Theories (GUTs) that link the strong, electromagnetic, and weak
interactions (Chapter 2): these are so widely accepted, at least in some fonn, as to be
practically orthodox. Also very attractive to theorists is the idea of 'supersymmetry'
(SUSY), which (among other things) would double the number of particles,
associating with every fennion a boson, and vice versa. Thus the leptons would
be joined by 'sleptons' (,selectrons', 'sneutrinos', etc.) and quarks by 'squarks';
the mediators would acquire twins (the 'photino', 'glun
i o', 'wino', and 'zn
i o').
If subquarks or supersymmetric particles are discovered, this will be huge news,
resetting thewholeagenda for the next era in elementary particle physics. But except
for several tantalizing false alarms [57], no evidence for either has yet appeared.
And then there is superstring theory. which since 1984 has captured the
imagination of an entire generation of particle theorists. Superstrings promise
not only to reconcile quantum m«hanics and general relativity, and to eliminate
the infinities that plague quantum field theory, but also to provide a unified 'theory
of everything', from which all of elementary particle physics (including gravity)
would emerge as an inescapable consequence. String theory has certainly enjoyed
a brilliant and adventurous youth; t
i remains to be seen whether it can deliver on
its extravagant ambition [58J.
References
1 There �re many good discussions
of the history of el",mcntary pu·
tide physics. My own f�VQrit'" is
th", ddightful little book by Yang,
C. N. (1961) fll:memary Par/ides.
Princeton University Press. Prince­
ton, N.J. More recent accounts are
(�) Trefil's, J. S. (1980) From AIO"'l
10 Quaru. Scribners, New York:
and (b) Clos",'s, F. E. (198]) n...
Cosmic Onion, Heinemann Educa·
tional Books, London_ The early days
are treated well n
i (c) Keller's. A.
(1983) The Infancy ofAtomic Physics.
Oxf(lrd University Press. Oxford;
and (d) Weinberg's. S. (1983) Sub·
atomic Particles. Scientific American
Ubrary, New York.For a fascinating and masterful aCC(lunt. see ("')
Pais. A.(1986) In\ll(lrd Bound. Claren­
don Press, Oxford. For a compTe­
h",nsive bibliography see (f) Hovis.
R. C. and Kragh, H. (1991) Amer­
I",n Jounwl of PhySICS> 59. 779.
1 The story is be�utifully told by
Pais. A_ in his biography of Ein­
stein. (1982) Sub/II: is 1M Lord,
Clarendon Press, Oxford.
I. T I The 5tandord Mod'" (/978-?)
3 Millihn. R. A. (1916) Physi.
cal Rnoiew, 7. 18. Quoted in
(�) Pais, A. in his biogr.lphy
of Einst�in. (1982) Subtlt is the
Lord. Cl�r�ndon Press, Oxford.
• H�isenberg had suggested ��rli�r
that th� deuteron is h�ld tog�th�r by
exchang� of dutTOns, in �n;tlogy with
th� hydrogen mole<:ul� ion (HiT
Yukawa was app,arently 1m first to
understand that a new force was in·
volved, distinct both from ele<:tro­
Jrulgnetism and from the weak force
r�sponsible for beta deciY. s� Pais,
A, (1986) Illword Bound, Clarendon
Press. Oxford; �nd (a) Brown. L. M.
and Re<:henberg. H. (1988) Amer·
jea,. Journal ofPhysi<;s. 56, 982.
5 Conv�rsi. M.• Pancini. E. �nd
Piccioni. O. (1947) PhJlSiwl R�.
71. 209.
6 Lattes. C. M. G. tl aI. (1947) Nature.
159. 694; (1947) Nalure. 160. 453.
.S<;.
7 Marsmk. R, E, and Belbe. H. A.
(1947) PhJlSiwl Review. n. 506.
Actually, lap,anese physicists md
come to a similar conclusion
during 1m war; � (a) Pais, A.
(1986) Illward Bound. Claren·
don Press. O�ford, p. "53,
8 For an infonnal history of this dis·
covery. see Anderson. C. D. (1961)
AmeriwlI Journal of PhJlSics. 29. 825.
9 Cmmberlain. O. tt aI. (1955) Phys·
iwl Review. 100, 947; (a) Cork, B.
,( al. (1956) Physical Review. 1(14.
1193. S� the articles by (b)
Segre. E. �d Wiegand, C. E.
(June 1956) Sciemijic Ameriwn.
37 �nd by (c) Burbridge. G. and
Hoyle, F. (April 1958), J-ot.. Also
(d) Kragh. H, (1989) A_rican
journal of Physics, 57. 10J-ot..
HI The history of the neutrino is a fas·
cinating story in its own right. See.
for instance, Bernstein. I. (2004) 1M
E1u.iVl! Neutrin<). University Press
of the Pacific, Honolulu; (a) Brown.
L M. (September 1978). Physics
Today, 23; or (b) Morrison. P. (lan·
uary 195-6) Sdenlifie A_ricall. 58.
An extensive and useful bibliogra·
phy on the neutrino is provided by
(c) Lederman, L. M. (1970) Amer·
iwn Journal of Physics. 38. 129.
11 Reines. F. and Cowan, C. L. Ir..
(1953) Physical Rnoiew, 92. 8301;
(i) Cowan. C. L et aI. (19S6) Sci·
'IIU. 1204. 103. Reines finally won
the Nobd Priu for this work in
1995 (Cowan was dead by then).
12 Davis, R. and Harmer, D. S. (1959)
Bulldi,. of1M American Physio­
kigical Society. 4. 217. See also
(a) Cowan. C. L Ir and Reines.
F. (1957) Physiwl Review. 1(16.
825.
13 Konopinski, E. I. and Mahmoud.
H. M. (1953) Physical Review. 92.
1045.
h Pontecorvo. B. (1960) Swid PhJlSUs
JErP. 37, 1236; (translated from
{l9S9} S""i.t Physi<;sJETP. 37. 1751);
(a) Lee. T. D. (1960) Rochester
Conference. New York. p. 567.
15 Danby. G. d aI. (19(2) Physi·
cal Rev"", uUm. 9. 36. See
also (a) Lederman, L (March
1963) 5<;ientiji� A_riwn, 60.
16 Brown. L. M Dresden M.
.•
.•
.
and Hoddeson, t. (November
1988). PhJlSics Today. 56.
(19..7) Nature, 160, 855. See
17 Rochester. G. D. and Butler. C. C.
also G. D. Rochester's memoir
in (a) Sekido. Y. and Elliot. H.
(eds) (l98S) Early Hirtory of Cos·
mi� Ray Studitl. Reidel. Dordrecht.
p. 299.
18 StUckelberg himself did not use
the term b�ryon. which was intro­
duced by Pais, A. (1953) Progress
ill Thwretical Physics. 10. "57.
19 Us Prix No«! 1955 The No­
bd Foumbtion. Stockholm.
20 Pais, A. (1952) Physical Rnoitlll. 86.
663. The nme (copious production,
slow deay) could b!' said for the
pion (and for that matter 1m neu·
tron). But their decays produce neu'
uinas. and people were used to ihe
idea that neutrino inter.lctions are
weak. What was new was a purdy
hadronic deay whose rate was char·
acteristic of neutrino proc�sses. For
more on the history see (a) Pais, A.
(1986) Inward Bo"nd, Clarendon
I 53
5-41 1 Hislorical lntroduc6oM to the fl�mento'l' Partid..
Pr"'ss, Oxford, p. 517: or (b) Brown.
L. M., Or�d",". M. �nd Hoddeson,
L. (Nov"'mkr 1988), PkyriQ Today,
60
21 Gdl·Mann, M. (1953) Pk�ical Re·
vUw, 92. 883: (a) (1956) NUDVO
CinuMto, 4 (Suppl. 2), 8-48.
22 Nakano, T. and Nishijima. K. (1953)
Progress iM Thuraical Pkysics. 10. 581.
23 Th", origin411 p;opers �n: col·
lected in Gell·Mann, M. and
Ne'em�n. Y. (1%4) The Eigkifold
Way, Benjamin, New York.
24 Actually. there is a possibility that
it was seen in a cosmic ray ""'peri­
ment in 1954 Eisenkrg, Y. (195-4)
Pkyrical Rl;lliew, 96, 541, but th",
identification was ambiguous.
2S Barnes. V. E. d aI. (1964) Pk)'$·
icaJ &\oj..., {dIm, 12, 20-4: (a)
Gell·Mann, M. (1953) Pk)'$"al R.·
vi...,. 92. 883. s..,e also (b) Fowler,
w. B. and S�mios, N. P. (Octokr
1%4), Scientific AmU'ican, 36.
26 An extensive bibliography on the
quark model, �nd useful commen·
tary, is gi�n by Gr"""nkrg. o. w.
(1982) AnuricaM journal of Pk)'$ics.
so. 107-4. M�ny of the classic pa·
pers (including th", original un·
published one by G. Z�ig) are
reprinted in (a) lichtenberg, D. B.
and Rosen, S. P. (eds) (1980) o.V<lop­
mtnts in the Quaoi: Theory of Hadrons.
Hadronic Press. Nonanrum.
27 Nambu. Y. (Novernkr 1976) S<:i·
entifi' Anurican, 48: (a) Johnson.
K. (July 1979) Sckntifi' Anuri·
caM. 112: (b) Rebbi, C. (Febru·
ary 1983) Scientific American,
S<
28 Kendall. H. W. and Panofsky.
W. K. H. (June 1971) S,ient'fi< Ameri.
can, 61.
29 Jacob, M. and Landshoff, P. (March
1980) Scknt'fi< American. 66.
10 Greenkrg. O. w. (1%4) Phy$ical
Review Ltuers, 13, 598. Green·
berg did not use this language:
the terminology w�s introduced by
(a) lichtenkrg, O. B. (1970) UMi.
tary Symnutry and BenuMttll)' Par·
ti�, Acac:\",mic Press, New York.
S""" �lso (h) Gluhow, S. L. (Oc·
tober 1975) ScieMtific AnuricaM.
3S.
31 S""" Pais. A. {1986} IMward Bound,
Clarendon Press, Oxford. pp. 562
and 602.
n Trilling, G. (2006) Revi..., of Partick
Pkysics. p. 1019.
n For st�tus reports see Bauer,
G. (2006) InternatioMl jour·
Mal ofModern Pkysics. A21. 959;
and (a) Swanson. E. S. {2006}
Physics Reports, 429, 2-43.
l( An exception was lliopoulos, J.
(197() AI all IMternaliol101 COMfirtll(;e
of Panick Physicists. London, in the
summer of 1974, he remarked, ·1
am n:ady to kt now � whole case
[ofwine) that the entire n""'t Con·
ferenc", wi
l l be dominatN by the
disco�ry of charmed particl�·
35 Aukrt, /. /. a aJ. (1974) Ph}'Sj·
cal Review LtUtrs. 33. 1-4()(; �nd
(a) Augustin. J.·E. et al. {19H} Pk)'$·
ical Rl;lli..., latm. 33, 1406.
16 A useful bibliography on this mate·
rial. and reprints of the major arti·
des. is given in Rosner, J. led) {1981}
New Pk)'$ics, published by tM Amer·
ican Association of Physics Teach·
ers. New York. The excitement of
the N"""'mkr R",volution is up­
tured in the SLAC publication; (a)
(1976) &am Lill<:, 7 {II}. S""" also
the �rtides by {b) Orell. S. o. (lune
1975) s<:ielllific Americall, 50, and
(e) Glashow, S. L. and Trilling. G.
(ZOO6) RPP 2006, p. 1019.
37 Bjorken, B. J. and Glashow. S. L.
(l964) Pkysicl Laters, 11, 255.
In 1963 and 1964 then: wen:
many speculations �bout a pos.
sible fourth quark, for � v�riety
of different r",asons (s""" Pais, A.
(1986) IMward BouMd. Claren·
don Press. Oid"ord. p. &01.
lS Glashow, S. L, lliopoulos, J. and
Maiani, L. {1970) Pk)'$kal Rev,...,. 02.
1285.
n Schwitters, R. F. (October 1977) Scj·
eMtific AmericaM. 56.
40 uz:roli. E. G. et al. {197S} Ph}'S·
ical Review Letters. )-4, 1125.
1.11 Tht Swndard
.1 Goldhaber, G. a aJ. (1976) Phys·
ical Review Letters, 37, 255: and
(a) Peruzzi, I. (1976) Physi.
cui Review utters, 37, 569.
.2 Brandelik, R. a aJ. Physics utttrs,
B70, 132.
.3 Perl. M. " al. (1975) Physiclll Review
Ltturs, 35, 1489. See also (a) Perl,
M. and Kirk, W. (March 1978) Sci·
tnl!fic Amtrican, SO: and (b) Perl.
M. (Cktober 1997) Physics Today. 3-4.
Perl was awarded the Nobel Prize
in 1995 for the discovery of the T.
.. Herb, S. W. tI "I. (1977) Physiwl
Rtvi.... Ltttm, 39. 252. See also
(a) Ledennan, l. M. (O<:tober 1978)
Sciuuific American, 72. [t is an indio
ution of how eager people were to
find the fifth quark that the discov·
ererS of the upsilon jumped the gun
(b) (Hom, D. C d aJ. (1976) Physic,,1
Review ultlln, 36, 1236), announc'
ing a spurious partid.. now known
fondly as the ·oops·l<"on' (after l<"on
.5 B..hrends, S. d <li. (1983) Ph)'S·
ad of th.. group).
Ledennan, h..
iwl Review ultm, SO, 881.
•
6 Abe, F. " <li. (1995) Physicui Revi....
LtJurs. 74, 2626:
and (a) Abachi.
S. d <li. (1995) Physic<>1 Rtview Ltt·
Ius, 74. 2632.
See also (b) !iss,
T. M. and Tipton, P. l. (Septem·
ber 1997), Scuntific Ameri""n.
"
.7 TIt.. formula for th.. W and Z
masses was first obtained by
utters, 19, 126•. [t involves a pa.ram·
W.-inberg. S. (1%7) Phyriml Rtv;'",
et<"r Ow whose valu.. was unknown at
that time. and all Wcinberg could say
for sure was that Mw ": 37 GeV{cl
and Mz � 75 GeV{rJ. In the next
15 years Ow was measured in a va·
riety of experiments, and by 1982
the predictions h.-id been refined,
as indicated in Equation ((1.30)).
.8 Amison, G. d al. (1983) Physics w·
Urs, 122B, 103.
d Amison, G. tt <li. (1983) Physics LtJ·
urs, 1268, 398.
W Cline, D. B. and Rubbi<o, C (Au­
gust 1980) Physics Today, 44: and
(a) Clin... D. B., Rubbia, C. and
Modd
(1978-?)
van der Meet, S. (March 1982) Sci·
tnt!fic Amtrican, 48. In 1984 Rubbia and van der Meer were awarded
the Nobel Prize for this work. See
(b) Sutton, C. (1984) The P"rtide
ConlltClion, Simon & Schust<"r, New
York; and (c) Watkins, P. (1986)
Story of the W "nd Z, Cambridge
University Press, Cambridge.
51 Jacob. M. and landshoff, P.
(March 1980) Scientific American,
66.
S2 Ishikawa. K. (November 1982), Sci·
entific Americ"n, a2: (a) Sexton, I·,
Vaccarino, A. and W..ingarten, D.
(1995) PhYfiwl Review uUm, 75,
4563.
S3 See, for instance, the review ar·
tide by Terez.awa, H. (1984)
XXII International Conjtrtl1Ce on
High·EM'1l' PhysiCJ, vol. I, J...-ipzig,
p. 63: and (a) Haran, H. (April
1983) Sdent!fic American, 56.
S. Cline, D. B. (August 1988) Scient!fic
American, 60.
SS Abrams, G. S. tt aI. (1989) Phyrical
Rev;'", wtt...., 63. 2173. See also (a)
I.
Feldman, G.
and Steinberger, I.
(F..bruary 1991), Scientific American,
70
56 For a delightfuJ discussion,
see Cahn, R. N. (1996) Revitws
ofModtrn Phl"ics, 68, 951.
57 On the search for subquarks, see
Abe, F. d aI. (1996) Physical Review
ulur$. '77, 5336. For the evapo­
rating evidence of supersymmetry
in measurements of the anomalous
magnetic mom..nt of th.. muon.
see (a) Schwarzschild, 8. (Febru·
�ry 2002) Phl"ics Today, 18.
ss For a brilliant semi·popular account,
see Greene, B. (1999) Tht Ekganl
Universe, W. W. Norton, New York.
For an accessible n
i troduction to
the theory, see (a) Zwiebach, B.
(2004) A Fir$t Course in SIring The·
ory, Cambridge, UK, Cambridge
University Press.) For harsh cri·
tiques see (b) Smolin, l. {2006)
The Trouble with Physics. Houghton
Mifflin, Boston; and (e) Woit, P.
(2006) Not Elltn Wrong, Perseus,
New York.
I
SS
S6
I
J HistoriUlI iMlroduaion 10 Ihe Elememo'Y Prlrticlel
Problems
\.I If � cru.rged p;trticle is undeflected in p;tssing through uniform crossed electric �nd
magn�tic fields E and B {murually perpendicular and both perpendicular to th� direction
of motion). what is its velocity� If w� now rum off the electric fi�ld. and the particle
moves in an arc of radius R. wru.t is its cru.rge-to-mass r;ltio�
1.2 The mass of yuk;!wa's meson can be estimated as follows, When two protons n
i a
nucleus achang� a meson {mass m), they must tempor.lrily violate the conservation of
energy by an amount "",2 {the rest energy of w meson), The Heis�nberg uncenainty
principle s�ys that you may 'borrow' �n energy 6£. provided you 'pay it back' in a time
6t given by fl.E 6t: fl/2 {wh�� fI .. hI21f), In this�, w� need to borrow fl.E: "",2
long enough for the meson to mak� it from one proton to the other, [t ru.s to cross the
nucleus (siz� '0), and it travels,presumably, at some substantial f!1lction ofthe speed of
light, so, roughly speaking, fl.t: role, Putting all this together, we have
h
m � -2roc
Using fO _ 10-11 cm {the size of a typical nucleus), c;Ucul�te the mass of yuk;!Wi.'S
m�son, Express your answer in MeVle2, and compare w observed mass of the pion,
(Commellt: If you find that argument compelling, I can only say th;ot you're pretty
gullible, Try it for an alOm, and you'll conclude that w mass of the photon is about
7 x IO-JOg, which is nonsense, Nevertheless, it is a useful device for 'back..,f-the·
envelope' cakulations, and it does very well for th� pi meson. Unfortunately, many
books present it as though it were a rigorous derivation, which it certainly is Mol. Th�
unceminty principle does Mol license violation of conserv�tion of energy (nor does
any such violation occur in this process; we shall see later on how this comes about).
Moreover, it's an illtljuality, fl.E fl.1 ::: /1(2, which at most could give you a k>wtr bound
on m. [t il typically troe that the rangt ofa force is inversely proportional to the mass of
the mediator, but the size of a bound sbte is not always a good measur� of the range.
(lbat's why the argument fails for the photon: th� rang� ofw electromagMtic force is
infinite, but the size ofan atom is not.) In general. when you hear a physicist invoke the
uncertainty principle, keep a hand on your wallet.!
1.3 In the period before the discovery ofthe neutron, many people thought th;ot the nucleus
consisted ofprotons and dulroMI, with the atomic number equal to w excess number
of protons. Beb deay seemed to suppon this idea - after all, �lectrons come popping
out; doesn't that imply th;ot there were electrons inside? Use th� position-momentum
uncertainty relation, fl." 6p 2! fi/2, to estimate the minimum momenNm ofan electron
confined to a nucleus (radius 1O-llcm). From the �Jativistic energy-momenrum
rel�tion, E2 p2el = m2c', determine the corresponding energy and comp;tr� it with
that of an electron emitted in, say, the beta deay of tritium (Figure 1.5). (Ibis result
convinced some people th;ot w beta deay electron could MOl have been rattling around
insid� th� nucleus, but must be produced in the disintegration itself.)
L4 The GeIl·Ma,.IIjOkubo mossformula relates the masses of members of the baryon octet
(ignoring small differences between p and M; 1: +, 1:°, and 1: -; and SO and S-):
_
Using this formula, together with the known masses ofw ",.atOll N (use w average
of p and II). 1: (again, use the average). and S (ditto), 'predict' th� mass of the A . How
close do you come to 1M obs�rved value?
J. J 1 Th� SlDndord Model
(1978-?) 157
1.5 The s�me formul� �ppJies to the mesons (with t ..... ", " ..... �, etc.), except that in this
Cilse, for reasons th;o.t rermtin something of a mystery, you must use the squares of the
masses. Use this to 'predict' the mass ofthe 1/. How close do you come?
1.{; The mass formula fordewplets is much simpler - equal sp..ong between the rows'
mll - mp = m:\:, - m:;:, = m:;:, - mn
Use this formul� (�s GeU·Mmn did) to predict the m�ss ofthe 0-. (Use the �verage of
the first two SP_lCings toestimate the third.) Howdose is your prediction to theobserved
VlIlue?
1.7 (a) Members of the b;oryon decuplet typically decay after lO-llseconds n
i to a lighter
b;oryon (from the baryon octet) and a meson (from the pseudo-scalar meson octet).
Thus. for ex�mple. ;l.H ..... p+ +
List all decay modes of this fonn for the ;l.- ,
to+, and 3'-. Remember that these decays must conserve charge and strangeness
,,+.
(they are strong interactions).
{hI In �ny decay, there must be sufficient m�ss in the original p,article to COlIer the
masses of the decay products. (There may be more than enough; the enra will be
'soaked up' in the form of kinetic energy in the final state.) Check each of the
decays you proposed in p�rt (�) to see which ones meet this criterion. The others are
kinematically forbidden.
1.& (a) Anal)'"le the possible decay modes ofthe 0-, just as you did in Problem 1.7 for the 6,
t.., �nd S*. See the problem? Gell·Mann predicted that the 0- would be 'metastable'
(i.e. much longer lived than the other members of the decuplet), for precisely this
re�son. (The 0- MeS in fact decay. but by the much slower weak interaction. which
does not conserve strangeness.)
{h) From the bubble chamber photograph (Figure 1.9), measure the length of the 0track, and use this to estimate the lifetime of the 0-. (Of course. you don't know
how fast it was going, hut it's � safe bet that the speed was less than the velocity
of light; let's say it was going about O.k Also, you don't know if the reproduction
has en!�rged or shrunk the scale. but never mind: this is quibbling over factors of 2.
or 5, or maybe even 10. The important point is that the lifetime s
i m�ny orders of
magnitude longer than the lO-IJ seconds characteristic ofall other members of the
deruplet).
1.� check. the CoIema,,-Glashow relation [Ph)'$. Rtv. 8134. 671 (1964)):
(the p;ortide Ill\mes stand for their rmtsses).
1.10 look up the table of 'known' mesons compiled by Roos. M. (1963) Rtviews ofMOIkrn
Phl"ics, 35, 314, �nd comp,are the cunent Panicle Physics Booklet to determine which of
the 1%3 mesons have stood the test of time. (Some ofthe ruomes h�ve been changed.
so you will have to work from other properties. such �s mass. charge. str�ngeness,
etc.)
1.11 Of the spurious p,articles you identified n
i Problem 1.10, which �re 'exotic' (I.e.,
inconsistent with the quark model)? How many of the surviving mesons are exotic?
1.12 How many different nuso" combinations can you make with 1. 2. 3. 4. 5, or (, different
q\Urk lbvors? What's the general formula for " flavors)
l.U How many different ""'i"''' combiIll\tions can you make with I. 2, 3. 4, 5, or (, different
quark flavors? What's the gener�1 formula for " flavors?
1.1.. Using four quarks (II, d, I, and c), construct � table of �ll the possible baryon species.
How many combinations carry a charm of +P How many carry charm +2, and
+3)
1.1S Same as Problem 1.14. but this time for mesons.
SI
I T Hi.loriwl InfrOduaion f<) Ihe EiemMIIlr,r Parficitl
1.16 Assuming the top qu�rk is too short·lived to form bound states (,truthful' mesons and
baryons), list the IS distinct meson combinations qij {not counting �ntipartides) and
the 35 distinct haryon combinations 9q9. From the Panicle Ph)'$ics Booklet and/or other
sources, determine which of these haw: been found experimentally. Give their name,
mass, and year ofdiscow:ry {just the lightest one, in each case)_ Thus, for instance, one
baryon entry would be
sss : Q-, 1672 MeV(rf,
1964.
All hl.drons are {presumably) various excitations ofth� SO quark combinations.
1.17 A. De Rujub, H. Georgi, and S. L. Glashow [PhysiCtJi Review, 012, 147 {1975l1 estimated
1M so·called conslitutnlqUllrk ma=" to be: mu = m� = 336 MeV/r, "" = 540 MeV/cI,
and In.: = 1500 MeV/�I {the bottom quark is about 4500 MeV/�I)_ Tfthey are right, the
average binding energy for members of the baryon octet is -62 MeV. If they all had
eXllGlly this binding energy, wh;tt would their masses be? Compare 1M ad",,) values
and giw: the percent error. {Don't try this on the other supennultiplets, however. There
really is no reason to suppose that the binding energy is the same for aU members of
the group. The problem of hadron masses is a thorny issue, to which we shall return in
Chapter 5.)
1.18 Shupe, M. (1979) [Ph)'$icr utlm, 86B, 87) proposed tmt all qu;orks and leptons are
composed of two even more elementary constituents: c {with (mrge -1/3) and II {with
charge zero) - and their respective antiparticles, ;; and n. You're allowed to combine
them in groups of three particles or three antiparticles {CCII, for example, or ;;/iii}.
Constructall ofthe eight qu;orks and leptons in 1M first generation in this Jrulnner. (The
other generations are supposed to be excited states.) Notice that each of the q""rl: states
admits three possible pennutations {UII, '""" nee, for example) - these correspond to the
three colors. Mediators un be constructed from three particles plus three antiparticles.
W±, zO, and r involve three Wee particles and three like antiparticles {W- = ccciiiiii,
for instance). Construct W.. , zO, and y in this way. Gluons involve mixed combinations
{""" ", for instance). How many possibilities �re there in �11� Can you think of any
way to reduce this down to eight?
1.19 Your roommate is a chemistry major. She knows all about protons, neutrons, and
elertrons, and she sees them in �ction every day in the laboratory. But she is skeptical
when you tell her about positrons, muons, neutrinos, pions, quarks. and intermediate
vector bosons. Explain to her why none ofthese plays any direct role in chemistry. (For
instance, inthe use ofthemuon a reasonable answer might be They are unstable, and
last only a millionth ofa second before disintegrating.')
course, the dfective mass of a quark bound inside � h..t.on
is not 1M �me �s the 'In.e' mass of the 'free' <lUlf\(.
• Foe reaSons _ wiD COme to in due
I"
2
Elementary Particle Dynamics
This chap�r introduces thefundalmnwl forces by which elementary particles interact,
and 1M Feynman diagroms WI: use to represent these interactions. 'The treatment is
entirely qualitative and can be read qlJicHy to gel a sense ofthe 'lay of 1M land'. Tht:
quantitative details will come in Chapters 6 through 9.
2.1
The Four Forces
As rar as we know, there are just four fundamental forces in nature: strong.
electromagnetic, wt:ak, and gravitatioll(ll. They are listed in the following table in
order ofdecreasing strength:'
Force
Strength
Strong
Electromagnetic
10
10-2
Weak
Gravitational
lO-IJ
10-.2
Th�'Y
Chromodynamics
Electrodynamics
Flavordynamics
Geometrodynamics
Medi ator
Gluon
photon
WandZ
Craviton
To each ofthese forces there belongs a physical theory. The classical theory ofgravity
is Einstein's general theory of relativity (,geometrodynamics' would be a better
is, of course, Newton's law of universal gravitation. Its relativistic generalization
term). A completely satisfactory quantum theory of gravity has yet to be worked
out; for the moment, most people assume thaI gravity is simply too weak to play
a significant role in elementary particle physics. The physical theory that describes
electromagnetic forces is called electrodynamics. It was givenits classical formulation
• The 'strength' of � force is �n intrinsk�11y ambiguous notion - aftM aU, i� dep"nds on the
n�ture of the source and on how far away you aI<:. So the numbers in this table should not
be taken too literally, and (especi"Uy in the Qse of the weak force) you will see quite different
figures quoted elsewhere.
ISBN, 978-H27-40601·1
Inlrod"u;"" 10 fltmtnl4ry P�rI;dts. Second Edition. D.vid Griffith.
Copyright C> 2008 W1LEY·VCH VulaS GmbH & Co. KGaA, Weinheim
60
I2
EI,mUIIOty Porticl, Dynumics
by Maxwell over one hundred years ago. Maxwell's theory was already consistent
with special relativity (for which it was, in fact, the main inspiration). The quantum
theory of electrodynamics was perfected by Tomonaga, Feynman, and Schwinger
in the 19405. Theweak forces, which account for nuclear beta decay (and also, as we
have seen, the decay ofthe pion, the muon, and many ofthe strange particles), were
unknown to classical physics; their theoretical description was given a relativistic
quantum formulation right from the start. The first theory of the weak forces
was presented by Fenni in 1933; it was refined by Lee and Yang, Feynman and
Gell-Mann, and many others, in the 19505, and put into its present form by Glashow,
Weinberg, and Salam, in the 19605. For reasons that will appear in due course,
the theory ofweak interactions is sometimes calledjlavcrdynamics {1J; in this book,
I refer to it simply as the Glashow-Weinberg-Salam (GWS) theory. (The GWS
model treats weak and electromagnetic interactions as different manifestations of
a single electroweak force, and in this sense the four forces reduce to three.) As for
the strong forces, beyond the pioneering work ofYukawa in 1934 there really was
no theory until the emergence ofchromodynamics n
i the 1970s.
Each of these forces is mediate<! by the exchange of a particle. The gravitational
forces are mediated by the graviton, electromagnetic forces are mediated by the
photon, strong forces by the gluon, and weak forces by the intermediate vector bosons,
W and Z. These mediators transmit the force between one quark or lepton and
another. In principle, the force of impact between a bat and a baseball is nothing
but the combined interaction of the quarks and leptons in one with the quarks
and leptons in the other. More to the point, the strong force between two protons,
say, which Yukawa took to be a fundamental and irreducible process, must be
regarded as a complicated interaction of six quarks. This is clearly not the place
to look for simplicity. Rather, we must begin by analyzing the force between one
truly elementary particle and another. In this chapter, I will show you qualitatively
how each ofthe relevant forces acts on n
i dividual quarks and leptons. Subsequent
chapters develop the machinery needed to make the theory quantitative.
2.2
Quantum Electrodynamics (QED)
Quantum electrodynamics (QED) is the oldest, the simplest, and the most sue·
cessful ofthe dynamical theories; the others are self-consciously modele<! on it. So
I'll begin with a description of QED. AI! electromagnetic phenomena are ultimately
rtducibk to thefollowing ekmentary process:
y
•
•
1.1 QIiO/1llim Ekr;lrod)'t'om;Cf (QED)
In these figures time flows horizontally, to the right, so this diagram reads: a
charged particle, e, enters, emits (or absorbs) a photon, y, and exits. For the sake of
argument, I'll assume that the charged particle is an electron; it could just as well
be a quark, or any lepton except a neutrino (the latter is neutral, ofcourse, and does
not experience an electromagnetic force).
To describe more complicated processes, we simply combine two or more repli·
cas of this primitive vertex. Imagine that you have a bag full of 'tinker toy' models
of the primitive vertex. made out of flexible plastic. You can snap them together,
photon-to-photon or electron-to-electron (but in the latter case you must preserve
the direction ofthe arrows). Consider, for example, the following:
,
,
y
,
,
Here, two electrons enter, a photon passes between them (I need not say which
one emits the photon and which one absorbs it; the diagram represents both
orderings), and the two exit.' This diagram, then, describes the interaction between
two electrons; in the classical theory, we would call it the Coulomb repulsion oflike
charges. In QED, this process is called Moller scall-tring; we say that the interaction
is 'mediated by the exchange ofa photon', for reasons that should now be apparent.
You're allowed to twist these 'Feynman diagrams' around into any topological
configuration you like - for example, we could stand the previous picture on its side:
,
,
y
,
,
A particle line running 'backward in time' (an arrow pointing toward the left) is
interpreted as the corresponding antiparticle going forward (the photon is its own
antiparticle, that's why I didn't nee<! an arrow on the photon line). In this process
• In ""ading a Feynman diagram it sometimes helps to picture • •ertical line that s""� alonS
to the right, ""presenting the passage of time. In the beginning (far left) it intersects two dec­
tron lines, in the middle it encounters the exchan� photon, and at the end (far right] there
are again just tw<J dectrons.
1 61
621 1 ElemtnUlry Particle Dyllamia
an electron and a positron' annihilate to form a photon, which in turn produces
a new electron-positron pair. An electron and a positron went ill, an electron and
a positron came out (not the same ones, but then, since all electrons are identical,
it hardly matters). lbis represents the interaction of two opposite charges: their
Coulomb attraction. In QED, this process is called Bhabha scattering. Actually, there
is a quite different diagram which also describes Bhabha scattering:
•
•
•
•
As we shall see, both diagrams must be included in the analysis.
Using just two vertices we can also construct the following diagrams, describing,
respectively, pair annihilation, e- + e+ _ y + y; pair production, y + Y _
e- + e+; and Compton scattering, e- + y -+ e- + y:
x
Notice that Bhabha and M0ller scattering are related by crossing symmetry
(Section 1.4), as are the three processes shown here. In tenns of Feynman di·
agrams, crossing symmetry corresponds to twisting or rotating the figure. Ifwe
allow more vertices (just reach in the bag and pull out a few mOre tinker toys), the
possibilities rapidly proliferate; for example, with four vertices we obtain, among
others, the follOwing diagrams:
, Some authoa W<)uld 1:o�1 the UP!"" left and
lower right lines in this diasnm with to 10 rr·
mind you Wol iI's an anti�rtic1e. I think Ihis
is dangerous notation. The arrow already tdls
you iI'S the anti�rtic1e. and a Uter� rrading
would suggeSI Wot il is an antipnlic1e going
backwards in time . . . which would � a par·
tid/:. ! prd"e' 10 la�l aU lines with Ihe parl;m
symbol. and let the arrow �ll you whether it
is in fact the anti�"icle.
2.2 Quan:..m flearodrrwmics (QED)
In each ofthese figures two electrons went in and two electrons came out. They
too describe the repulsion of like charges (Moller scattering). The 'innards' of the
diagram are irrelevant as far as the obselVed process is concerned, Internal lines
(those which begin and end within the diagram) represent particles that are not
obselVed - indeed, that cannot be obselVed without entirely changing the process.
We call them virtl«il p�rticles. Only the i'-Xkm�l lines (those that enter or leave
the diagram) represent 'real' (obselVable) particles. The external lines, then, tell
you what physical process is occurring: the internal lines describe the mtCh�nism
involved.
At the purely qualitative level this is such a childishly simple game that there's
a serious danger you will inadvertently embel!ish the rules. If you find yourself
drawing a Feynman diagram that contains the vertex
for example, or
x
or you snap a photon line onto an eie<:tron line
you have made � mistake
-
the bag contains no such tinker toys, and the snaps just
don't work when you try to hook a photon to an el«tron. Your diagram might
conceivably describe some otka interaction, but it's not ele<:trodynamics.
163
641 2 ElemenlOry Palticie Dy�omjcs
Feynman diagrams are purely symbolic; they do notrepresent particle trajectories
(as you might see them in, say, a bubble chamber photograph). The horizontal
dimension is time, but vertical spacing does llet correspond to physical separation.
For instance, in Bhabha scattering the electron and positron are attmcl-td, not
repelled (as the diverging lines might seem to suggest). All that the diagram says
is: 'Once there was an electron and a positron: they exchanged a photon: then there
was an electron and a positron again'.
Quantitatively, each Feynman diagram stands for a particular lIumba, which
can be calculated using the so-called F�ynmall ntks (you'U learn how to do this
in Chapter 6). Suppose you want to analY.le a certain physical process (say,
M011er scattering). First you draw all the diagrams that have the appropriate
external lines (the one with two vertices. all the ones with four vertices, and
so on), then you evaluate the contribution of each diagram. using the Feyn­
man rules, and add it all up. The sum total of all Feynman diagrams with the
given external lines represents the actual physical process. of course, there's a
wee problem here: there are infinitely many Feynman diagrams for any partie·
ular reaction! Fortunately, each vertex within a diagram introduces a factor of
Q = �21fu; = 1/137, theJiIl� stl1lcture COl1Stallt. Because this is such a small num·
ber, diagrams with more and more vertices contribute less and less to the final
result, and. depending on the accuracy you need, may be ignored. In fact, in
QED it is rare to see a calculation that includes diagrams with more than four
vertices. The answers are only approximate, to be sure, but when the approx­
imation is valid to six significant digits, only the most fastidious are likely to
complain.
The Feynman rules enforce conservation of energy and momentum at each
vertex, and hence for the diagram as a whole. It follows that the primitive QED
vertex by itself does not represent a possible physical process. We can draw
the diagram. but calculation would assign to it the number z�ro. The reason is
purely kinematical: �- -+ e- + y would violate conservation of energy. (In the
center-of-mass frame the electron is initially at rest, so its energy is mel. It cannot
decay into a photon plus a recoiling electron because the latter alone would require
an energy greater than me2.) Nor, for instance, is e- + e+ -+ y kinematically
possible, although it is easy enough to draw the diagram:
,
,
y
In the center-of·mass system the electron and positron enter symmetrically with
equal and opposite velocities, so the total momentum before the collision is
obviously zero. But the Jinal momentum callnot be zero, since photons always
22 Quanlum Eieclrodynamics (QED)
trave! at the speed of light; an electron-positron pair can annihilate to make two
photons, but not one. Within a larger diagram, however, these figures are perfectly
acceptable, because, although energy and momentum must be (Onserved at each
vertex, a virtllal parfick � not wIT)' the salm mass as the corresponding free
particle. In fact, a virtual particle can have any mass." In the business, we say that
virtual particles do not lie on their mass sht:U. External lines, by contrast, represent
rtal particles, and these do carry the 'correct' mass.t
I have been assuming that the charged particle in question s
i an electron,t but it
(ould just as well be a muon, say, or a quark. What would you make ofthe following
diagram?
u
u
u
Here a Il(ii pair annihilates, producing two photons (one photon, remember, is
kinematically forbidden). Because of quark confinement you're not going to wit­
ness this as a scattering experiment, but what If the quarks were bound together
in the form of a meson - a ]1"0, for example? This diagram would represent the
'decay' ofthe ]1"0: ]1"0 -+ y + y. I put the word in quotes, because in a deeper sense
this is not a decay at all - it's lust ordinary old pair annihilation, in which the
original pair happen to be bound together as a meson. This explains why the ]1"0
has a lifetime 9 orders of magnitude smaller than its charged siblings (]I"±) - it
decays by an electromagnetic process, whereas the others have to await the weak
interactions, which are much slower.
I cannot resist telling you an amusing fable, but you must promise not to take
it too seriously. Feynman claimed that his advisor (/. A. Wheeler) once offered the
a photon from a distant star would have to
be ext�mely close to its 'correct' mass - it
tum. p. and mass m of a fr� p<.rticle are �.
would have to be alllrl<>l ·rea\". As a
latN by the �uation E' _ p1cl .. m'c'. BUI
tional man..", you would get �ssentially the
for a virtual p<.rtide E' - p'c' an take on any
....me answ.." if)'<lu treat� the process as two
valu�. Many authors inlerp�t this to mean
$epUat� events (emission of a real photon by
tlul virtual proceues viola.le (onservation of
star, followed by ibsorption of a �al photon
�n�rgy (� Problem 1.1). Pnsonally, [ Con·
by t�). You might ....y that a real p<.rtide is a
sider this miduding, at besl. Energy s
i ,,1....1"
.
virtual partic� that la.sts long tnough that we
conserved.
don't ca� to inquire how it was produced. or
t Actually. the physiul distinction between �al
how it is eventuaUy absorbed.
and virtual p<.rtides is not quite as shup as
I hive impliN. 1{ a photon is emitted on AI· t In practice, the term 'quantum electrodynam·
ics' is usually taken to mean the interaction
pha Centauri, and absorbed in )'<lur eye. it is
of electrons. positrons, and photons, unl=
tecbnially a virtu./.l pholon. t suppose. How.
otheIWise sp«ified.
ever, in g�neral, the futher a virtual puticle
is from its mus shell the shorter it lives. sO
• [n 5�1 rdati...;!)'. th� '''''''1P' E, mom�n·
alcula.·
1 65
661 2 Elementary Parljde DytlamjQ'
It's riding along on a da
i gram of the form
following explanation for why all electrons are identical: there's only o� of 'em!
times as a particle and three times as an antiparticle - but it's all the same electron,
At a given instant (the vertical line) the electron is present (on this segment) four
Of course, this does imply that the number of positrons in the universe should
equal the number ofelectrons (give or take one), but apart from that it's kind ofcute.
2.3
Quantum Chromodynamics (QCD)
In chromodynamics, cclor plays the role of charge, and the fundamental process
*
(analogous to e --+ e + y) is quark --+ quark plus gluon (q --+ q + g):
,
q
q
As before, we combine two or more such 'primitive vertices' to represent more
complicated processes. For example, the force between two quarks (which s
i
responsible in the first instance for binding quarks together to make hadrons,
and indirectly for holding the neutrons and protons together to form a nucleus) is
described in lowest order by the diagram:
q
q
,
q
q
* Sin� leptons do not u.rry color. they do not �rticipo.te in the strons interactions.
2.1 Quan/L1m Chromodyll<lmjcs (QeD)
We say that the force between two quarks is 'mediated' by the exchange of
gluons.
At this level chromodynamics is very similar to electrodynamics. However, there
are also important differences, most conspicuously the fact that whereas there is
only one kind ofelectric charge (it can be positive or negative. to be sure, but a single
color (red, green, and blue). In the fundamental process q � q + g, the color ofthe
number suffices to characterize the charge of a particle), there are three kinds of
quark (but not its flavor) may change. For example, a blue up-quark may convert
into a red up.quark. Since color (like charge) is always conserved, this means that
the gluon must carry away the difference - in this instance, one unit of blueness
and minus one unit ofredness:
g(b,'"
u(b)
u(r)
Gluons. then, are 'bicolored', carrying one positive unit of color and one nega­
tive unit. There are evidently
3 x3
=
9 possibilities here, and you might expect
there to be nine kinds of gluons. For technical reasons, which we'll come to in
Chapter 8, there are actually only eight.
Since the gluons themselves carry color (unlike the photon, which is electrically
neutral), they couple directly to other gluons, and hence in addition to the
fundamental quark-gluon vertex, we also have primitive gluon-gluon vertices; in
fact, two kinds: three·gluon vertices and four-gluon vertices:
rX
This direct gluon-gluon coupling makes chromodynamics a lot more complicated
than electrodynamics, but also far richer, allowing, for instance, the possibility of
glueballs (bound states ofinteracting gluons, with no quarks in Sight).
Another difference between chromodynamics and electrodynamics is the size
of the coupling constant. Remember that each vertex in QED introduces a factor
of ct = 1/137, and the smallness of this number means that we need only
consider Feynman diagrams with a small number of vertices. Experimentally, the
corresponding coupling constant for the strong forces, ct, - as determine<!, say,
from the force between two protons - is greater than 1, and the bigness of this
number has plagued particle physics for decades. Instead of contributing less
and less, the more complex diagrams contribute more and more, and Feynman's
procedure, which worked so well in QED, s
i apparently doomed. One of the great
161
681 2
Elemer1!Qrv Particle ay""mics
•
''-.
I:
'
�
•
,
••
-
/'
�
,
'.
I
.I , � ,
,
Fig, 2,1 Scrt<!fling of a charge q by a ditlectr;c medium,
triumphs ofquantum chromodynamics (QCD) was the discovery that in this theory
the number that plays the role of coupling 'constant' is in fact not constant at
all, but depends on the separation distance between the interacting particles (we
call it a 'running' coupling constant), Although at the relatively large distances
characteristic of nuclear physics it is big, at very short distances (less than the size
of a proton) it becomes quite small, This phenomenon is known as �symptOlic
freedom [2]; it means that within a proton or a pion, say, the quarks rattle around
without interacting much, Just such behavior was found experimentally in the deep
inelastic scattering experiments. From a theoretical point of view, the discovery of
asymptotic freedom rescued the Feynman calculus as a legitimate tool for QCD, in
the high·energy regime.
Even in electrodynamics, the effective coupling depends somewhat on how far
you are from the source. This can be understood qualitatively as follows. Pictwe first
a positive point charge q embedded in a dielectric medium (i.e. a substance whose
molecules become polarized in the presence of an electric field). The negative end
of each molecular dipole will be attracted toward q, and the positive end repelled
away, as shown in Figure 2.1 As a result, the particle acquires a 'halo' of negative
charge that partially cancels its field. In the presence of the dielectric, then, the
effective charge of any particle is somewhat reduced:
qeff = q/f
(2.1)
i reduced is called the ditltctric constant of the
(The factor E by which the field s
material; it is a measure of the ease with which the substance can be polarized
[3].) Of course, if you are closer than the nearest molecule, then there s
i no such
screening, and you 'see' the full charge q. Thus if you were to make a graph of the
2,) QII(i"t�m Chromody"omics (QCD)
q
q
,
- - - --""-""r----
,
Intermole<:ular
5eparation
Fig. 2.2 Effective ch;rge ;s ; function of disunce.
effective charge, as a function of distance, it would look something like Figure 2.2
The effective charge increases at very small distances.
Now, it so happens that in quantum electrodynamics the vacuum itself behaves
like a dielectric; it sprouts positron-electron pairs, as shown in Feynman diagrams
such as these:
...etc.
The virtual electron in each 'bubble' s
i attracted toward q. and the virtual positron
and reduces its field. Once again, however, ifyou get too close to q, the screening
is repelled away; the resulting vacuum polarization partially screens the charge
Compton wavelength of the electron, .l.., = hjrnc = 2.43 x 10-10 cm. For distances
disappears. What plays the role of the 'intennolecular spacing' in this case is the
smaller than this the effective charge increases, just as it did in Figure 2,2.
Notice that the unscretmd (,dose.up·) charge, which you might regard as the
'true' charge of the particle, is not what we measure in any ordinary experiment,
1 69
70
I 2 fI�m�Mtory Porfjde Oynomje.
since we are seldom working at such minute separation distances.' What we have
always called 'the charge of the electron' is actually the fully screened effective
charge.
So much for electrodynamics. The same thing happens in QeD, but with an
important added ingredient. Not only do we have the quark-quark-gluon vertex
(which, by itself, would again lead to an increasing coupling strength at short
distances), but now there are also the direct gluon-gluon vertices. So in addition to
the diagrams analogous to vacuum polarization in QED, we must now also include
gluon loops, such as these:
It is not clear a priori what influence these diagrams will have on the story (4J; as
it turns out, their effe<t s
i the opposi�: There occurs a kind of competition between
the quark polarization diagrams (which drive as up at short distances) and gluon
polarization (which drives it down). Since the fonner depends on the number of
on the number of gluons (hence on the number of colo�, II), the winner in the
quarks in the theory (hence on the number ofjkIVO�,f), whereas the latter depends
competition depends on the relative number of flavors and colors. The critical
parameter turns out to be
a := 2f - lln
(2.2)
!fthis number is positive, then, as in QED, the effective coupling increast:S at short
II :::: 3, so a = -21, and the QeD coupling decreases at short distances. This is the
distances; if it is lIegative, the couping
l
dureaus. In the Standard Model,f = 6 and
origin ofasymptotic freedom.
The final distinction between QED and QeD is that whereas many particles carry
electric charge. no naturally occurring particles carry color. Quarks are confined
in colorless packages of two (mesons) and three (baryons). As a consequence,
the processes we actually observe in the laboratory are necessarily indirect and
complicated manifestations of chromodynamics. It is as though our only access to
electrodynamics came from the van der Waals forces between neutral molecules.
For example, the (strong) force between two protons involves (among many others)
• An exception ;s the umb shifl - a tiny �rl>ation in the spe<trum of hydrogen - in which
the influence of vacuum polarization (or rather, il5 "b"'Mct at shon distances) is dearly dis_
cernible.
2.4 We.>K /ntemctieru
the following diagram:
(P)
d
"
"
"
"
"
"
(P)
(p)
d
"
d
(I (Jt")
"
d
"
(p)
You will recognize here the remnants ofYukawa's pion-exchange model. but the
entire process is enormously more complex than Yukawa ever imagined.
If QeD is correct, it must contain the explanation for quark confinement; that
is, it must be possible to provt, as a consequence of this theory, that quarks can
only exist in the form of colorless combinations. Presumably this proofwill take
the fonn of a demonstration that the potential energy increases without limit as
the quarks are pulled farther and farther apart, so that it would require an infinite
energy (or at any rate, enough to create new quark-antiquark pairs) to separate
them completely (see Figure 2.3). So far, no one has provided a conclusive proof
that QeD implies confinement (see, however, Reference 27 in Chapter 1). The
difficulty is that confinement involves the Iong-mnge behavior of the quark-quark
interaction, but this is precisely the regime in which the Feynman calculus fails:
2.'
Weak Interactions
There is no particular name for the 'stuff that produces weak forces, in the sense
that electric charge produces electromagnetic forces and color produces strong
• There are streng indic.lIiens Wt � 'phase
Ir.I.nsition' cccurs at extremdy high densi·
ties _ th� er feur times that of an a!emie
the so·c.olled q""rt_g/uclI pl<uma. Thus, f�
nucleus - leading Ie deconfinement �nd
quarks may have aisted in the first mements
after the Big Bang. and elfe"" are underway
to recrule similar conditions (en a SJrnUer
sc.o]el) in the laboratery, using the Relativistic
Heavy len CoUider (RH1C) at Brookh�ven (5).
1 71
72] 2
Elementary Partide �l1(JmjGS
<==
<==
<==
8 8 0 =:>
'-,-'
,
0 === 0 0=== 0 0 =:>
00
�
.'
Fig. 2.3 A possible seen�r;o for qu�rk confinement: �s we
puff � u qu�rk out of the proton. � p,air of qu�rks is ereated.
and instud of a fr� qu�rk. we ue left with � pion �nd �
neutron.
forces. Some people call it 'weak charge'. Whatever word you use, all quarks and all
leptons carry it (6J. (Leptons have no color, so they do not participate in the strong
interactions; neutrinos have no charge, so they experience no electromagnetic
forces; but aU of them join in the weak interactions.) There are two kinds of weak
interactions: charged (mediated by the Ws) and �utral (mediated by the 2). The
neutral weak interactions are much simpler, so I'll start with them:
2.4.1
Neutral
The fundamental neutral vertex is:t
z
where f can be any lepton or any quark. The Z mediates such processes as
neutrino-ele.::tron scattering (v" + e- -+ v" + e-):
•
Although charg�d w�ak m
i �ra(tions w�r�
known risht from the stan (beta deca� is the
classic uampl�). th� theoretical possibility of
neutral we�k processes was not �ppreciated
until 1958. The GWS model indudes neutr�l
_ak inteucrions as essenml ingredients.
and th�ir uL.tenc� waS fint (onfirmtd in
neutrino scattering exptrimtnts �t CERN. in
1973 (7).
t Itis traditional to use a wavy lint for thephoton,
and a springy line for the gluon. but there is
no consistency in the literaturt f<lr the weak
medial",s. rm SrunS 1<1 use a jagged line, but
this is n<lt standard n<>tati<ln. {I'll use .. solid
lin� for spin·1/2 particles, which is standard.
and a dashed line for spin O. which is n<lq
2.4 Weok !nleroclrons
z
"
"
and neutrino-proton scattering (v" + p --+ v" + p)
z
d
d
::::(p)
::=.-tfr--=::.::::
(P)
(in the latter case, two 'spectator' quarks go along for the ride, bound to the d by
strong forces - gluon exchange - that, for simplicity, we do not draw)_'
Notice that any process mediated by the photon could also be mediated by the
Z - for example, ele<:lron-electron scattering:
e
,
e
e
z
"
Presumably there is a minute correction to Coulomb's law attributable to the
se<:ond diagram, but the photon-mediated process overwhelmingly dominates.
Experiments at DESY (in Hamburg) studied the reaction e- + e+ --+ 1-'- + 1-'+ at
very high energy and found unmistakable evidence of a contribution from the Z
(S]. In atomic physics, neutral weak contamination of electromagnetic processes
can sometimes be teased out by exploiting the fact that weak interactions carry
a unique fingerprint: they violate conservation of parity (mirror symmetry) (9).
1 73
7<1
I2
fI�m�"lQry Particle Dynllmics
Still. to observe a pure neutral weak interaction one has to resort to neutrino
scattering. in which there is no competing electromagnetic mechanism - and
neutrino experiments are notoriously difficult.
2.4.2
Charged
The primitive vertices for strong. electromagnetic. and neutral weak interactions
all share the feature that the same quark or lepton comes out as went in - accompa·
nie<!. ofcourse. by a gluon. photon. or Z. as the case may be. Well . . . OK: in QeD
the color of the quark may change. but the flavor never does. The charged weak
interactions are the only ones that change flavor. and in this sense they are the only
ones capable of causing a 'true' decay (as opposed to a mere repackaging of the
quarks. or a hidden pair production or annihilation). I'll begin with the charged
weak interactions ofleptons.'
2.4.2.1
Leptons
The fundamental charged vertex looks like this:
neutrino. with emission of a W- (or absorption of a W+): [- --+ VI + W-.t As
A negative lepton (it could be e- . /1-- . or f-) converts n
i to the corresponding
always. we combine the primitive vertices to generate more complicated reactions.
For example. the process /1- + v. --+ e- + v I' would be represented by the
diagram:
W'
Such a neutrino-muon scattering event would be hard to set up in the laboratory.
but with a slight twist essentially the same diagram describes the decay of the
• The disco� of neutrino oscillation. will force some modification. in this pkrure. but we do
not know yd e><Icdy what form they will t.ake (perhaps they will bring the theory into line with
the quarks). SO for the moment I will stick to the simple (pre-<>scillation) story.
t This implies. of course. th.at the nOMed reaction 1+ .... ;;1 + w+ is also allowed.
2.4 Weak Intuactions
muon, �- _ e- + V" + Ii,:
,
.
"
w-
This is the cleanest of all charged weak interactions; we'll study it in detail in
Chapter 10:
2.4.3
Quarks
Notice that the leptonic weak vertices connect members ofthe same gentration: e
converts to V, (with emission of W-). or �- _ �- (emitti ng a 2), bute- never goes
to �- nor �- to Vt. In this way, the theory enforces the conselVation of electron
number, muon number, and tau number. It is tempting to suppose that the same
rule applies to the quarks, so that the fundamental charged vertex is:
W'
A quark with charge - � (which is to say: d, s, or b) converts into the corresponding
(u, c, or t, respectively), with the emission of a W-. The
quark with
charge +�
outgoing quark carries the same color as the ingoing one, but a different flavor.'
The far end of the W line can couple to leptons (a 'semileptonic' process), or to
other quarks (a purely hadronic process). The most important semileptonic process
is a + v. _ u + e:
.
.
"
W'
d
"
• Technically, it is only th� lowrst-order contribution to muon decay. but in weak interaction the­
t It's not that the W- carries off the 'minins· Haver
ory one almost ne....r �s to ,omider hisher.()r<kr corr�tiom.
_ the WOs h""" no flam,; flavor is simply ""I
,on..rv«I in the chargN weak interactions.
1 75
761 2 ElemtnUlry Particle DynllmiC$
Because of quark confinement, this process would never occur in nature as it
stands. However, turned on its side, and with the ii and d bound together (by the
strong force), this diagram represents a possible decay of the pion, rr - -+ �- + Ii,:
"
,
•
d
(For reasons to be discussed tater, the more common decay is actually rr - -+
1-1- + iiI" but the diagram is the same, with t replaced by 1-1-.) Moreover, essentially
the same diagram accounts for the beta decay of the neutron (n � p+ + e- +
iie):
•
W·
d
Thus, apart from strong interaction contamination (in the form of the spectator
quarks). the decay of the neutron is identical in structure to the decay of the muon.
and closely related to the decay of the pion. In the days before the quark mode\,
these appeared to be three very different processes.
Eliminating the electron-neutrino vertex in favor of a second quark vertex we
obtain a purely hadronic weak interaction, /::,.0 -+ p+ + rr-:'
d
w
d
d
• Th� lJ.O ha5 the ume quark «mlent as the neutron. but this deay is not possible for neutron5
because they are not heavy enough to makr • proton and a pion.
2.4 W�(l'k 1111�f(l'ctio"5
Actually, this particular decay also proceeds by the strong interaction:
(.,
d
"
g
d
"
d
W)
�E-�(p)
the weak met:hanism s
i an immeasurably small contribution. We'll see more
realistic examples of nonleptonic weak interactions in a moment,
So far, it's all pretty simple: the quarks mimic the leptons - the only difference
is that the strong force (to which, remember, the leptons are immune) complicates
the picture with sp«tators and what-not that have nothing to do with the basic weak
process. Sad to say, this story is a little too simple. For as long as the fundamental
quark vertex is allowed to operate only within roch generation, we can never hope
to account for strangeness-.changing weak interactions, such as the decay of the
lambda (A -+ p+ + ]1"-) or the omega.minus (Q- -+ A + K-), which involves the
conversion of a strange quark into an up quark:
d
w
(A)
,
(K')
"
"
::::I:=-k P)
lbe solution to this dilemma was suggeste<l by Cabibbo in 1963, perf«ted
by Glashow, lIliopoulos, and Maiani (GIM) in 1970, and extended to three
generations by Kobayashi and Maskawa (KM) in 1973.' The essential idea is
that the quark generations are 'skewed.' for the purposes of weak interactions.t
• The Cabibbo/GIM/KM m@�nism wiU tit
discUS5W mo� fully in Chapt�r 9.
t T@nicall�, this applies 10 the neutral as
�ll as th� charged w�ak int�ractions. But in
the fonner case it doesn't nuner. and I ha�
tr� to k�p th� story as dear as possible by
avoiding the issue at that s�ge. Historically.
when thae wa� only thr� quarks known
it WaS a puzzle why (experimen�lly) thae
wae no strangeness·chansing neutral weak
n
i �ractions. The GIM m...:hani$m introduced
a fourth quark (fout �ar5 IItfort the Novern·
btor Revolution), and a 2 x 2 'KM matrill', to
provide for a miraculous canceUation. the net
df...:t ofwhich (in th� n�utraJ cas�) wu the
same as if we had never 'skewed' the quarks
n
i the first place,
I
n
18
I
2 Elementa')' Particle Dynamic.
Instead of
(2.3)
the weak force couples the pairs
(2.4)
where d', s, and b' are linear combinations ofthe physical quarks d, s, and b:
(�) (�� �: �:) (:)
b'
=
VoJ
v"
v,.
(2.5)
b
If this 3 x 3 Kobayashi-MaskaJrn matrix were the unit matrix, then d', S, and b'
would be the same as d, s, and b, and no 'cross·generational' transitions could
occur. 'Upness-plus..aownness' would be absolutely conserved (just as the electron
number is); 'strangeness-plus-charm' would be conserved (like muon number);
and so would 'topness-plus-bottomness' (like tau number). But it's nol the unit
matrix (although it's pretty dose); experimentally, the magnitudes of the matrix
elements are [lOJ
(�:��; �:!�� �::)
0.008
0.042
(2.6)
0.999
Vud measures the couping
l of u to d, V�, the coupling of u to s, and so on. The fact
that the latter is nonzero is what permits strangeness-changing processes, such as
the decay ofthe and the n-, to occur.'
A
2.4.4
Weak and Electromagnetic Couplings of W and Z
There are also direct couplings of W and Z to one another, in GWS theory (just as
there are direct gluon-gluon couplings in QeD):
�z � Vz
� � �
• Neutrino oscil!.1otion5 invol"., cross·�nerati0lt.11 couplings in the kp!on SKlor, so it ma� be WI
we will h�ve a 'KM Imtrix' for the leplons as well. See C1upter 11.
2.5 OIi'roys �nd ConstlV�lion LDws
Moreover, because the W is charged, it couples to the photon:
I, � X,
� � :
Although these interactions are critical for the internal consistency of the theory,
they are oflimited practical importance at this stage (see Problem 2.6).
2.'
Decays and Conservation Laws
One of the most striking general properties of elementary particles is their
tendency to disintegrate; we might almost call it a universal principle that �v�ty
particle duays into lighter particles, unlessprevtntedfrom dcing so by some const1V(.llion
law. The photon is stable (having zero mass, there is nothing lighter for it to
decay into); the electron is stable (it's the lightest charged particle, so conservation
of charge prevents its decay); the proton is presumably stable (it's the lightest
baryon, and the conservation of baryon number saves it - likewise conservation
of lepton number protects the lightest of the neutrinos). By the same token, the
positron, the antiproton, and the lightest antineutrino are stable. But most particles
spontaneously disintegrate - even the neutron, although it becomes stable in the
protective environment of many atomic nuclei. In practice, our world is populated
mainly by protons, neutrons, electrons, photons, and neutrinos; more exotic things
are created now and then (by collisions) but they don't last long. Each unstable
species has a characteristic mean lifetime,' r : for the muon it's 2.2 x 10-� se<:; for
it's 8.3 x 10- 17 sec. In fact, most particles
the
it's 2.6 X 10-8 sec; for the
exhibit several different decay modes; 64% of all
for example, de<:ay into
�+ + III" but 21% go to]l"+ + ".0, 6%to".+ + ]1"+ +]1" -, 5% to (e + v, + ]1"0), and
so on. One of the goals of elementary particle theory is to calculate these lifetimes
and branching ratios.
A given decay is governed by one of the three fundamental forces: t::.++
.
-+
p+ + "'+, for example, s
i a strong decay; ./to -+ y + y is electromagnetic; and
L- -+ n + e- + ii, is weak. How can you tell? Well, ifa photon comes out, the pro­
cess is certainly ele<:tromagnetic, and ifa neutrino emerges, the process s
i certainly
weak. But if neither a photon nor a neutrino is present, it's a little harder to say.
For example, L- -+ n + ]I"- is weak. but t::.. -+ n + ]I" - is strong. I'll show you
in a moment how to figure that out, but first I want to mention the most dramatic
experimental difference between strong, electromagnetic, and weak decays: a typical
]1"+
]1"0
K+'s,
+
• TIw: lifotime � is related to the hrUf�ift '1/1 by the formula '1/1 "" �n2)T '"' 0.693T. The half·life
is the time il tokes for half Ihe �l1icles in a I.arSe S<lmple 10 disint.-grale (s� Section 6.1).
1 79
80
I2
ElemMlury Purticle Dy1lum;CJ
strong decay involves a lifetime around 10-23 sec, a typical electromagnetic decay
takes about 10-16 sec, and weak decay times range from around 1O-il sec (for
the r) up to 15 min (for the neutron). For a given type of interaction, the decay
generally proceeds more rapidly the larger the mass difference between the original
particle and the decay products, just as a ball rolls faster down a steeper hill.' It
is this kinematic effect that accounts for the enormous range in weak interaction
lifetimes. In particular, the proton and electron together are so close to the neutron's
mass that the decay n -+ p+ + e- + ii, b(lrdy makes it at all. and the lifetime ofthe
neutron is greater by far than that of any other unstable particle. Experimentally,
though, there is a vast separation in lifetime between strong and electromagnetic
decays (a factor of about 10 million), and again between electromagnetic and weak
decays (a factor of at least a thousand). Indeed, particle physicists are so used to
thinking in tenns of 10-23 sec as the 'normal' unit of time that the handbooks
generally classify anything with a lifetime greater than 10-17 sec or so as a 'stable'
particle!t
Now, what about the conservation laws which, as I say, permit certain reactions
and forbid others? To begin with there are the purely kinematic conservation
laws - conservation ofenergy and momentum (which we shall study in Chapter 3)
and conservation ofangular momentum (which comes n
i Chapter 4). The fact that
a particle cannot spontaneously decay into particles heavier than itself is actually
a consequence of conservation of energy (although it may seem so 'obvious' as
to require no explanation at all). The kinematic conservation laws apply to all
interactions - strong, electromagnetic, weak, and for that matter anything else
that may come along in the future - since they derive from special relativity itself.
However, our concern right now is with the dynamiwlconservation laws that follow
from the structure of the fundamental vertices:
q
z,w
y
9
q
,
• There are exceptions: If+ .... 1'+ + "�, for
example. is shorter by a factor of 10' than If+
..... e+ + "•. but such cases cry oUI for some
sped>.! explanation.
t incidentally, 1O-1J se.: is about the time it
takes light to cross a proton (diameter 10->5 mI. You obviously annot detennn
ie
the lifetime of such a particle with a stop·
watch, or even by measurins the Iensth of
its track (as you did for the 0- in Problem
1.8{b)) - it doesn't move far enousJ! to leQlIt a
Iud. T"-'lead, you make a hislosram of II"I4S$
,
r
me""u�ments, and n
i voke the uncertainty
principle: t>.Etll � /i/2. Here l>.E = (l>.",)2,
and t>.1 = T, so
•
p ---
- 2(6.m)c2
Thus the lprtad in I>IIm is a meaSUre of the
particle's lifetime. rredmia.!ly it's only a
I"""" bound on r. but for such short·lived
plrticles we are presumably risht up against
the uncertainty limit (II]).
2.5 OIcays ond Constrvation Lo""
Since all physical processes are obtained by sticking these together in elaborate
combinations, anything that is conserved at each vertex must be conserved for the
reactions as a whole. So, what do we have?
1. Cha'Ke: All three interactions, ofcourse, conserve electric
charge. In the case orthe weak interactions, the lepton (or
quark) that comes outmay not have the same charge as the one
thatwent in, but ifso, the difference is carried away by the W.
2. Color: The electromagnetic and weak interactions do not
affect color. At a strong vertex the quark color does change,
but the difference is carried offby the gluon. (The direct
gluon-gluon couplings also conserve color.) However. since
naturally occurring particles are always colorless, the
observable manifestation of color conservation is pretty
trivial: zero in, zero out.
3. Baryon m.mbu: In all the primitive vertices, ifa quark goes in,
a quark comes out, so the total number ofquarks present is
a constant. In this arithmetic we count antiquarks as negative,
so that, for example, at the vertex q + q -j. g the quark
number is zero before and zero after. Ofcourse, we never
see individual quarks, only baryons (with quark number
3), antibaryons (quark number 3) and mesons (quark
number zero). So, in practice, it is more convenient 10 speak
of the conservation of baryon number (1 for baryons, -1 for
antibaryons, and 0 for everything else). The baryon number
is just the quark number. Notice that there is no analogous
conservation of meson number; since mesonscarryzero quark
number, a given collision or decay can produce as many
mesons as it likes, consistent with conservation ofenergy.
4. upton number: The strong forces do not touch leptons
at all; in an electromagnetic interaction the same particle
comes out (accompanied by a photon) as went in; and in
the weak interactions if a lepton goes in, a lepton comes out
(not necessarily the same one, this time). So, lepton number
is absolutely conserved. Until recently there appeared
to be no cross.generation mixing among the leptons,
so electron number, muon number, and tau number were
all separately conserved. This remains true in most cases,
but neutrino oscillations indicate that it is not absolute:
-
,
t
• Th�r� would br a similar
conservalion ofgeneration type
for quarks (up�s·plus-downness,
stnngeness·plus..;:harm. and
b�au!)'·plus,'ruth), but h�r� th� inl�rg�n·
�rational mixing has been obvious for
decades. Still. brcau"" the off..:liagonal el·
em=!S in the KM matrix a", r�lati""ly small,
oO$s-generatiolllll decays lend to br sup­
crossings a", m",meiy ra", - hence an old
press...!. and pro«s""s thaI require IWO such
rule """ 'forbids' decays with 6S = 2.
I
al
82 [ 2 Elemenlary Particle Dynamics
5. Flavor: What about quarkflavor? Flavor is conselVed at a
strong or electromagnetic vertex, but not at a weak vertex,
where an up.quark may turn into a down quark or a strange
quark, with nothing at aU picking up the lost upness or
supplying the 'gained' downness or strangeness, Because the
weak forces are so weak, we say that the various flavors are
approximately conselVed. in fact, as you may remember, it
was precisely this approximate conselVation that led
Gell-Mann to introduce the notion of strangeness in the first
place. He 'explained' the fact that strange particles are always
produced in pairs:
(2.7)
for instance, but
(2.8)
by arguing that the latter violates conselVation of strange­
ness. (Actually, this is a possible weak interaction, but it will
never be seen in the laboratory, because it must compete
against enormously more probable strong processes that do
conselVe strangeness.) In dtcays, however, the
nonconselVation of strangeness is very conspicuous, because
for many particles this s
i the only way they can decay; there is
no competition from strong or electromagnetic processes.
The A, for instance, is the lightest strange baryon; if it is to
decay, it must go to n (or p) plus something. But the lightest
strange meson is the K, and n (or p) plus Kweighs
substantially more than the A. Ifthe A decays at all (and it
does as we know: A _ p+ + ]f- 64% ofthe time, and A --+ n
+ ]fo 3 6% ofthe time), then strangeness cannot be
conselVed, and the reaction must proceed by the weak
interaction. By contrast, the l::.0 (with a strangeness of zero)
can go to p+ + 7f or n + ]fo by the strong interaction, and its
lifetime is accordingly much shorter.
6. The 021 rule: Finally, I must tell you about
one very peculiar case that has been on my conscience since
Chapter L I have in mind the decay of the 1/1 meson, which,
you will recall, is a bound state ofthe charmed quark and its
antiquark: 1/1 = 'c. The 1/1 has an anomalously long lifetime
(�10-�O sec); the question is, why? It has nothing to do with
-
2.5 Decoys ond COnUMlI;ol1 Low.
conservation of charm; the net chann of the 1/1 is zero. The
1/1 lifetime is short enough so that the decay is clearly due to
the strong interactions. But why is it a thousand times slower
than a strong deuy 'ought' to be? The explanation (ifyou can
call it that) goes back to an old observation by Okubo. Zweig.
and Iizuka. known as the 'OZI rule'. These authors [12) were
puzzled by the fact that the q, meson (whose quark content,
is, makes it the strange analog to the 1/1) decays much more
often into two K's than into three H'S (the two pion decay is
forbidden for other reasons. which we will come to in Chapter
4). in spite ofthe fact that the three-pion decay is energetically
favored (the mass oftwo K's is 990 MeV/,2; three H'S weigh
only 415 MeV/,l). In Figure 2.4, we see that the three-pion
diagram can be cut in two by snipping only gluon lines.
The OZI rule states that such processes are 'suppressed'. Not
absolutely forbidden. mind you. for the decay q, ...... 3H does
in fact occur, but far less likely than one would otherwise have
supposed. The OZI rule is related to asymptotic freedom. in
the following sense: in an OZI.suppressed diagram the gluons
must b e 'hard' (high energy), since they carry the energy
necessary to make the hadrons n
i to which they fragment.
But asymptotic freedom says that gluons couple weakly
at high energies (short ranges). By contrast. in OZI-allowed
processes the gluons are typically 'soft' (low energy).
and in this regime the coupling is strong. Qualitatively. at
'J,
. {:
•
K-
. {:
•
.J
*
K-
Fig. 2.• The OZI rule: if the diagram can be cut in two by
�Iking only gluon lines (and not cutting any extern,,1 lines),
the process is suppressed.
•
•
•
:} .-
1113
841 2
E'tmemory Pr"1ic/e Dynomics
least. this accounts for the OZI rule. (The quantitative details
will have to await a more complete understanding of QCD.)
But what does all this have to do with the ",? Well.
presumably the same rule applies. suppressing '" ...... 31f. and
leaving the decay n
i to two charmed D mesons (analogs to the
K, but with the charmed quarks in place of the strange
quarks) as the favored route. Only there's a new twist in the
'" system, for the D's turn out to be too heavy: a pair of D's
weighs more than the "'. So the decay ", _ D+ + D- (or
rfJ +
15°) kilUmatically
is
forbidden, while", ...... 31f is 021
suppressed, and it is to this happy combination that the '"
owes its unusual longevity.
2.'
Unification Schemes
At one time. electricity and magnetism were two distinct subjects. the one dealing
and the North Pole. But n
i 1820 Oersted noticed that an electric current could
with pith balls. batteries. and lightning; the other with lodestones. bar magnets,
deflect a magnetic compass needle. and 10 years later Faraday discovered that
a moving magnet could generate an electric current in a nearby loop of wire.
By the time Maxwell put the whole theory together in its final form, electric·
ity and magnetism were properly regarded as two aspects of a single subject:
electromagnetism.
Einstein dreamed ofgoing a step further, combining gravity with electrodynamics
in a single unified jidd I�Ory. Although this program was not successful. a similar
vision inspired Glashow. Weinberg, and Salam to join the weak and electromagnetic
forces. Their theory starts out with four massless mediators, but. as it develops.
and the 2, while one remains massless: the photon. Although experimentally a
three ofthem acquire mass (by the so-called Higgs mechanism). becoming the W's
reaction mediated by W or 2 s
i quite different from one mediated by the y. they are
ooth manifestations of a single elearoweak interaction. The relative weakness of the
weak force is attributable to the enormous mass ofthe intermediate vector bosons;
its intrinsic strength is in fact somewhat greater than that of the electromagnetic
force. as we shall see n
i Chapter 9.
Beginning in the early 1970s, many people have worked on the obvious next step:
combining the strong force (chromodynamics) with the electroweak force (GWS).
Several different schemes for implementing this grand unification are now on the
table, and although it is too soon to draw any definitive conclusions, the basic idea
is widely accepted. You will recall that the strong coupling constant a, dureases at
short distances (which is to say, for very high-energy collisions). So too does the
weak coupling a.... but at a slower rate. Meanwhile, the electromagnetic coupling
constant, ae. which is the smallest of the three. increasts. Could it be that they all
converge to a common limiting value, at extremely high energy (Figure 2.5)? Such
2.6 unificatjon SC/!M"le5
10'� GeV
E
Fig. 2.S Evolution of th� thr� fimdam�nul coupling consunts.
is the promise of the grand unified theories (GUTs). Indeed, from the functional
form ofthe running coupling constants it is possible to estimate the energy at which
this unification occurs: around lOiS GeV. This is, of course, astronomically higher
than any currently accessible energy (remember, the mass ofthe Z is 90 GeVtel).
Nevertheless, it is an exciting idea, for it means that the observed difference in
strength among the three interactions is an 'accident' resulting from the fact that
we are obliged to work at low energies, where the unity of the forces is obscured.
If we could just get in close enough to see the 'true' strong, electric, and weak
charges, without any of the screening effects of vacuum polarization, we would
find that they are all equal. How nice!
Another prediction ofthe GUTs is that the proton is unstable, although its half-life
is fantastically long (at least 1019 times the age of the universe). It has often been
remarked that conservation ofcharge and color are in a sense more 'fundamental'
than the conservation of baryon number and lepton number, because charge is the
'source' for electrodynamics, and color for chromodynamics. If these quantities
were not conserved, QED and QeD would have to be completely reformulated. But
baryon number and lepton number do not function as sources for any interaction,
and their conservation has no deep dynamical significance. In the grand unified
theories new interactions are contemplated, permitting decays such as
(2.9)
in which baryon number and lepton number change. Several major experiments
have searched for these rare proton decays, but so far the results are negative [131.
If grand unification works, all of elementary particle physics will be reduced
to the action of a single force. The final step, then, will be to bring in gravity,
vindicating at last Einstein's dream, with the ultimate unification. At this point
superstring theory is the most promising approach.· Stay tuned!
• See Section 12.2 for more on grand unifiOltion, �nd Section 12.� for $Upwlymmetry and super·
strings.
I
as
86 1 Z
fltmtntory Pot1;id� t>.,(lQmics
References
1 Consis!ent etymology would call
for gtllsidyl'lami.;$, from the Greek
word for 'flavor': see Gaillard,
M. (April 1981) Physics Today, 74.
M. Gaillard suggests II5Iktl'lodymlm.
ics. from the Creek word for ,.,.,ak.
z Gross, PoHtzer. and Wilcek won the
2004 Nobel Prize for the discovery
of asymptolic freedom. For an a,:.
cessible account see Cross. O. J.
(January 1987) Physics Today. 39.
) See, for example, Purcell. E. M.
( 1985) Electricity al'ld Magmtism,
2nd cdn, McCraw·Hill, New York,
Sec. 10.1.
• Quigg, C. (April 1985) Scia;.
t!fie American. 84. giv<"S a Qual·
itative interpretation of the ef·
fect of gluon polarization.
s For a status report, see S. Aron­
son and T. Ludlam (2005) Hunt­
ing the Quark GIllon Plasma,
BNL-73847 BrookhaYl!n Na·
tional Labantory. Brookhaven.
6 The classk papers on weak interac·
lion tlu:ory up to 1%0 are collected
in Kabir. P. K. led) (1%3) The �­
!.'dopmtl'll of Weak Il'IIe"",twl'I The­
ory, Gordon & Breach. New York. A
similar collection covering the mod­
em era s
i contained in (a) !..ai, C. H.
(ed) (1981) Gaugt Theery of Weak and
E1Wromllgl1tti<: II'I�""'IWM, World
Scientific. Singapore. See also (b)
Commins, E. D, and Bucksb;!um.
P. H. (1983) Wtak Il'Iteractiol15 of up10115 al'ld Quam. Cambridge Uni­
versity Press. Cambridge, UK.
7 Hasert, F. J. tt aL (1973) Physics La­
Itn. 46B, 138; and N�r Physics.
B73. 1. See ;uso (a) Cline, D. B.,
Mann. A. K. and Rubbia, C. (Decem­
�r 1974) Sciel'll!fie Americal'l, 108.
S Wu, S. L (19M) Physics Reports,
107. 59), Section 5.6, See also
(a) Bouchiat, M. -A. and Pottier, L
(June 1984) Sciel'llifjc American. 100.
9 See Levi. B. C . (April 199'7) Pkysics
Today, 17.
10 The numbers are from the Par­
ticle Physics Bookld. (2006).
11 See Gillespie, D. T. (1973) A
Quallium Mtdlanics Primer. In·
!emalio""l Textbook Co., lon­
don, p. 78, for a careful justi­
fication of this procedure.
12 Olmbo, S. (1963) Pkysi<:s utlers, S, 16S; (a)Zweig. G. (1964)
CERN Preprints TH 401 �nd TH
411; (b)!izuka. J. (1%6) Progm!
il'l Physics S..ppI., 37, 21.
n Weinberg, S. (June 1981) Scien.t!fie
Atm:rical'l, 64; (a) LoSecco. J. M..
Reines, F. and Sinclair, O. (June
1985) Scientific American. S4. The
best current limits on the proton life·
time come from Su�r-K2miok.ande;
see (b)Shiozawa, M. tt al (1998)
Physicoll Rtview ul�", 81. 3319.
Problems
2.1 Calculate the ratio of the gravitational attraction to the electrical repulsion between two
stationary electrons. (Do ! need to tell you how far apart they arel)
Feynman diagram representing Dtlbruck scalrall'lg: y +
i classical
y ..... y + y. ("This process, the scattering of light by light, has no arulog o
electrodynamics.)
2.l Draw aU the fourth-order (folU Yl!rtex) diagr�ms for Compton scattering. ("There are 17
ofthem; disconnected diagrams doo't count.)
H Determine the mass of the virtual photon in each of the lowest-order diagrams for
Bhabha scattering (assume the electron and positron are at rest). What is its velocity)
(Note tlut these answers would be impossible for real photons.)
2.5 (0) Which decay do you think would be more likely,
2.2 Sketch the lowest-order
S- -> A + )"f-
or
S- -> n + rr-
Explain your answer, and coofirm it by looking up the ex�rimental data.
(b) Which deay ofUu, oO(cii) meson is most likely,
Which is leasl likely? Draw Uu, Feynman diagrams, 6plain your answer and check
the 6perimental data. (One of the successful predictions of the ubibboJGIMJKM
modd was that charmed mesons should decay preferentially into str.lnge mesons,
even though e"l''8'''t' ically the 2n mode is favored.)
Ie) How about the ·beautiful' (B) mesons? Should they go to Uu, D's. K's. or n·s?
2.6 Draw all the lowest-order diagrams contributing to the process e+ + e- ..... W+ + W-.
(One ofUu,m involves the direct coupling of Z to W's and another the coupling of y to
W·s. so when LEP (the electron-positron collider at CERN) achie\led sufficient energy
i 1996. these exotic processes could be studied experimentally. See
to make two W's, n
B. Schwarzsdlild, Physic$ Today (September 1996). p. 21.)
2.7 Examine the following processes, and state for each one whether it is possible Or
impa,sible, according to the Standard Modd (which does not include GUTs. with Uu,ir
potential violation of the conservation of lepton number and baryon number). In the
former case, state which n
i teraction is responsible - strong, electromagnetic, or weak;
in the latter case, cite a conservation law that prevents it from occurring: (Following
the usual custom. I will nol indicate the charge when it is unambiguous, thus y. A. and
n are neutral; p is positive. e is negative; etc.)
(e)
(a) P + P _ JT + +/T'
1;0 _ A +JTo
(e) e+ + t- _ � + + �­
(g) t.+ _ P + /To
(i) e + p _ u,+JTo
(k) p _ e+ + y
(m) It + ii _ JT+ + JT- + /To
(0) K- _ /T- + JTo
(q) EO _ A + y
(5) So _ P +JT(u) /To _ y + y
(b) ,, _ y + y
(d) E- _ n + JT­
(f)/J.- _ e- + v.
(h) v. + p _ n + e+
(j) p + p _ E+ + It + � + JT + + /T o
(I) p + p _ p + p + p + p
(n) /T+ + n _ JT - + P
(p) "E+ + n _ E- +p
(r) g- _ A + JT(I) JT- + P _ A + J<!l
(v) E- _ n + t +v.
2.8 Some decays n
i volve two (or even all three) different forces. Draw possible Feynman
diagrams for the following processes:
(I) Ji ..... t + e+e+ +u,,+D",
(h) l:+ ..... p + y
What interactions ue involved> (Both these decays have bel:n observed. by the
way.)
• Note: A wIIl,wn is never "inQnaticaliy forbid·
den. If)'Qu claim, for enmpk. that reaction
Ie) is forbidden by conservation of energy
(because the elKtron weighs less th..n Ihe
muon). )'Qu ..re at least half WTOI"Ig - it can
(and does) occur. as long as the electrons
t.av., enough kinetk energy to m.;oke up the
difference. But don·t try to pby this game for
duays a single particle cannal drca.y into
heavier secondaries no m.;oner ",hal its kinetic
energy is, as you em easily see by examining
the process in the rest fra"", of the decaying
_
particle.
as
I
2 flcmentQ'Y Porticle Ofoomin
The T meson, bb, is the bonom-quark analog to the 'It, d. Its mass is 9460 MeV/cl. and
its lifetime is 1.5 x 10-20 sec. From this information, what can you say about the mass
of the B meson. ub� (The observed mass is 5180 MeVfcl.)
l.IO The 'It' meson. at 3686 MeV/,l, h.as the same quark content as the t (viz. d). Its
principal decay mode is 'It' ..... 'It + If+ + If-. Is this a strong interaction? Is it
OZI suppressed� What lifetime would you expect for the t? [Ibe observed v;llue is
2.9
3 x lO-zl sec.)
2.11 Figure 1.9 shows the first confirmed production of an n-. in a hydrogen bubble
i cident K- evidently hit a station.try particle X. producing a �. a K+.
chamber. The n
and the n-. (a) What was the charge ofthe Xl Wlut was its strangeness? Wh..t particle do
you suppose itwas? (b) Follow each line in the right·hand di..gram.listing every reaction
as you go along; ;lisa specify what kind of interaction - {strong. electromagnetic. or
weak - WaS responsible. {In case the diagr.lm is undear. the two photons are supposed
to come from the same point. lncident;llly, while y ..... e- + c+ is impossible in vacuum
(it doesn't conserve momentum), it does occur in the vicinity ofa nucleus - the nucleus
SO<lks up the 'missing' momentum. The reaction is really y + p ..... e- + c+ + p, but the
p leaves nO track, because it is so heavy that it scarcely moves; the electron and positron
carry off the photon's energy. and the proton simply acts as a passive momentum
·sink'.)
2.12 The W- was discovered in 1983 at CERN, using proton/antiproton scattering:
p+p ...... W-+X
where X represents one or more particles. Wh..t is the most likely X. for this process�
Dr.lwa Feynman diagram for your reaction. and explain why your X is more proJnble
than the various alternatives.
I"
3
Relativistic Kinematics
In this ckapur, I summarize the basic principks, notation, and knmnology ofrdativisti<:
kinematics. This is material you must know cold in orlkr to understand Chapttrs 6
through 10 (it is not nudtdfor Chapters 4 and 5, howel.'tr, and ifyouprefer you can read
tkmfirst). Although the treatment s
i reasonably sdj-wntaiJUd, I do assume that you
kave encountered sptcial relativity before - if not, you should paust hut and read the
appropriate chapter in any introductory physics text before proceeding. Ifyou are already
quitt familiar with relativity, this chapter will Ix an easy review - but read through it
anyway buause some ofthe notation may be new to you.
3.1
Lorent:t Transformations
According to the spet:ial theory of relativity [IJ, the laws of physics apply just as
well in a reference system moving at constant velocity as they do in one at rest. An
embarrassing implication of this is that there's no way of telling which system (if
any) is at rest, and hence there is no way ofknowing what 'the' velocity ofany other
system might be. So perhaps [ had better start over. Ahem.
According to the spedal theory ofrelativity [l], thelaws ofphysics are equally valid
in all inutialreference systems. An inertial system is onein which Newton's first law
unless acted upon by some force: It's easy to see that any two inertial systems must
(the law ofinertia) is obeyed: objet:ts keep moving in straight lines at constant speeds
be moving at constant velocity with respet:t to one another, and conversely, that any
system moving at constant velOCitywith respect to an n
i ertial system is itselfinertial.
Imagine, then, that we have two inertial frames, S and 5', with 5' moving at
uniform velocity v (magnitude v) with respect to S (S, then, is moving at velocity
v with respect to 5'). We may as well lay out our coordinates in such a way that
-
the motion is along the common xIx! axis (Figure 3.1), and set the master docks at
= t' = 0 when x = x! = 0). Suppose, now, that some event occurs at position
the origin in each system so that both read zero at the instant the two coincide (that
is, t
• [f you are wondering whether � freely f�lling system in � uniform gr�vitational field is ·inertial'.
you know m� than i. good for you. �'. iuS! �p gravity out of it.
IllIrod"",iorI.lO S.tlltnlolry P,,,,;deJ. Swottd Edil;"'" David Griffiths
Copyright 0 2008 WILEY·VCH Verbg GmbH & Co. KG>A. W�inJ>..im
ISBN, 97&.3·527·-40601·1
90
I 3 Rtlalivislic Kintmalics
S
Y
S'
Y
,
Fig. 3.1 The inerti�1 systems 5 and 5'
(x, y, z) and time I in S. Q�lion: What are the space-time coordinates (x', y', z') and
( of this SQrnl event in S'? The answer is provided by the Lorentz transformations:
1. x' = y(x- tit)
i1. y'
=y
ii1. z = z
(3.1)
iv. t' = y (t - �x)
where
(3.2)
The inverse transformations, which take us back from S' to 5, are obtained by
simply changing the sign of tI (see Problem 3.1):
,
,
. x = y(x' + tit')
ii'. , � y
.. .,
lll . z = z
(J.J)
iv'. I = y (I' + �x')
The Lorentz transformations have a number of immediate consequences, of
which I mention briefly the most important:
1. Tht relativity ofsimultaruity: If two events occur at the same
time in S, but at different locations, then they do nol occur at
the same time in S. Specifically, if IA = ts, then
(3.4)
3. J Lorenlz Tramformalions
(see Problem 3.2). Events that are simultaneous in one
inertial system, then, are not simultaneous in others.
2. Lorentz contraction: Suppose a stick lies on the x' axis, at rest
in S'. Say one end is at the origin (x' "" 0) and the other is at
L' (so its length in S' is L'). What is its length as measured in
5? Since the stick is moving with respect to 5, we must be
careful to record the positions of its two ends at the same
instant, say t "" O. At that moment, the left end is at x 0 and
the right end, according to Equation (i), is at x = L'/y. Thus
the length of the stick is L "" L'/y, in 5. Notice that y is
always greater than or equal to 1. It follows that a moving
i shortenm by a factor of y, as compared with its length
object s
in the system in which it is at rest. Notice that Lorentz
contraction only applies to lengths along the direction of
motion; perpendicular dimensions are not affected.
3. Time dilation; Suppose the clock at the origin in S' ticks off
an interval 1"; for Simplicity, say it runs from t' "" 0 to t = 1".
How long is this period as measured in S? Well, it begins
when t "" 0, and it ends when t "" 1"' at x' 0, so (according
to Equation (iv')) t = y1". Evidently the docks in 5 tick off a
longer nterval,
i
T y 1", by that same factor of y; or, put it
the other way around; moving c/IXks run slow.
Unlike Lorentz contraction, which is only indirectly
relevant to elementary particle physics, time dilation is a
commonplace in the laboratory. For, in a sense, every
unstable particle has a built·in clock: whatever it is that tells
the particle when its time is up. And these internal clocks do
indeed run slow when the particle is moving. That is to say, a
moving particle lasts longer (by a factor of y) than it would at
rest." (The tabulated lifetimes are, of course, for particles at
rest.) In fact, the cosmic ray muons produced in the upper
atmosphere would never make it to ground level were it not
for time dilation (see Problem 3.4).
4. Velocity addition: Suppose a particle is moving in the x
dire<tion at speed u', with respect to S. What is its speed, u,
with respect to Sf Well, it travels a distance ax = y(ax' + v
at) in a time at = y[at + (v/el)ax'], so
=
=
=
ax
l\t
-
�
ax' + v at
at + (v/c2)l\x'
I
(ax'/t.f) + v
+ (v/c2)(l\x'/61')'
• Actu�ll�. the disintegration of an individu�l �rticle il � r..ndom process: when we spe�k of a
'lif�i�' we r�ally mean th� a...,"V lifetime of that partid� type. Wh�n I ny that a moving par·
ticle !;>.t. longer. I really mean th�t tile avt"'gt lifetime of � group of mavins �Tticles is lonser.
191
921 J RelOli_islic Kinemoli�
But 6x/6t = 1.1, and 6x/6t' = uf, so
"�
uf +v
;lc+�c("
""C/C";;1
(3.5)
Notice that if 1.1' = C, then u = C also: tht spud of light s
i the same in all inertia!
systems.
It can sometimes be confusing to figure out in a particular context, which
numbers should be primed and what signs attach to the velocities, so I personally
remember three rules: moving sticks are short (by a factor of y), moving docks
are slow (by a factor of y) - so put the y (which, remember, s
i greater than 1) on
whichever side ofthe equation you need to achieve these results, - and
(3.6)
where VAS (for instance) means thevelocity ofA with respect to B. The numerator is
the classical result (the So<alled 'Galilean velocity addition rule'); the denominator
is Einstein's correction - it is very close to 1 unless the velocities are close to c.
'.2
Four·vectors
[t s
i convenient at this point to introduce some simplifying notation. We define the
position.timefour.vector x". ;I. = 0, 1. 2, 3, as follows:
xl- = y,
,f = ct,
(3.7)
In terms of x" , the Lorentz transformations take on a more symmetrical appear·
ance:
l
O'
x = y(xo - px )
Xlf = Y (XI pxo)
x2' = xl_
(3.8)
where
(3.9)
More compactly:
(;I. = 0,},2,3)
(3.10)
3.2 Four-v«f<)1l
The coefficients A� may be regarded as the elements ofa matrix A:
-yp
y
o
o
o
o
(3.11)
1
o
(i.e. Ag = Al = y, A� = A? = -yfJ ; Ai = A� = 1; and all the rest are zero). To
avoid writing lots ofI:'s, we shall follow Einstein's 'summation convention', which
says that repeated Greek indices (one as subscript, one as superscript) are to be
summed from 0 to 3. Thus Equation 3.10 becomes, finally:
(3.12)
A special virtue of this tidy notation is that the same form describes Lorentz
transformations that are not along the
x direction; in fact, the Sand !1 axes need not
even be parallel; the A matrix is more complicated, naturally, but Equation 3.12 still
holds. (On the other hand, there s
i no real loss ofgenerality in using Equation 3.11,
since we are always free to choose parallel axes, and to align the
direction of v.)
x axis along the
Although the individual coordinates of an event change, n
i accordance with
Equation 3.12, when we go from S to $', there is a particular combination of them
that remains the same (Problem 3.8):
(3.13)
invariant. (In the same sense, the quantity ,2 = xl + r + z2 is invariant under
rotations.) Now, I would like to write this invariant in the form ofa sum: I:!.oxl-<xl-<,
Such a quantity, which has the same value in any inertial system, is called an
but unfortunately there are those three irritating minus signs. To keep track of
them, we introduce the mt:tnc, gl-'. ' whose components can be displayed as a
matrix g:
o
-1
o
o
o
o
-1
o
:]
(3.14)
-1
• In an expression such as this th� Greek
letl�r used for th� summation indrx, �. is
ofcourse compl�tely arbitrary. Th� Soi�
goes for th� 'hanging· indrx /l. although
it must match on the two sides of th�
equation. Thus Equation (3.12) could just
as wdl bo WIitt�n x" "" A�xI-. Either expres·
sion stands for th� ..., of four ..quation.:
./ = A�,!I + A:X' + Aixl + AJx'
xi = A�xo + A;x' + A�x2 + A�xl
Xl' = A�XO + A:X' + ,,�.? + "lx)
193
" I (i.e.
J Re/o/jvinic I(jnemoti<:l
goo = 1; gl1 = gll = gll = -1; all the rest are zero).' With the help of gIl'"� the
i
t I can be written as a double sum:
invaran
,
,
1 = L L&<.x"x· = 81" x"x"
(3.1S)
Carrying things a step further. we define the covariant four-vector X!, (index down)
as follows:
(3.16)
(i.e. Xo = ,,0, XI = -xl , Xl = xl Xl = _xl ). To emphasize the distinction we call
the 'original' four-vector x" (index up) a contmvariant four-vector. The invariant I
can then be written in its cleanest form:
-
,
(3.17)
(or, equivalently, x"x,,). All this wi
l l no doubt seem like monstrous notational
overkill, just to keep track ofthree pesky minus signs, but it's actually very simple,
once you get used to it. (What's more, it generalizes nicely to non-Cartesian
coordinate systems and to the curved spaces encountered in general relativity,
though neither ofthese is relevant to us here.)
The position.time four·vector x" is the archetype for allfour·vectors. We define
a four·vector, a", as a four·component object that transforms in the same way xl'
does when we go from one inertial system to another, to wit:
aI" = A�a·
(3.18)
with the same coefficients A�. To each such (contravariant) four·vector we associate
a covariant four·vector aI" obtained by simply changing the signs of the spatial
components, or, more formally
(3.19)
Of course. we can go back from covariant to contravariant by reversing the signs
again:
(3.20)
where g"V are technically the elements in the matrix g-I (however, since our metric
is its own inverse, g". is the same as gl'.)' Given any two four·vectors, a" and hI',
the quantity
(3.21)
• I should warn you that some physicists define the metric with the opposite signs (_I. I. I. 1). It
�n't maIler much - if 1 is an invlriant. so too is -1. But it �s mean Ih<It you must br On
the lookout for unfamiliar signs. Fortunately, most particle physicists nowadays use the con.....n·
lion in Equation 3.1•.
3.2 Four-ll.ao"
is invariant (the same number in any inertial system). We shall refer to it as the
scalar prodl<Ct of a and b; it is the four-dimensional analog to the dot product oftwo
three-vectors (there is no four-vector analog to the cross product).'
If you get tired ofwriting indices, feel free to use the dot notation:
a , b ... a,,/T'
(3.22)
However, you will then need a way to distinguish this four-dimensional scalar
product from the ordinary dot product of two three-vectors. The best way is to
be scrupulously careful to put an arrow over all three-vectors (except perhaps the
velocity, v, which, since it is not part of a four,vector, is not subject to ambiguity).
In this book, I use boldface for three-vectors. Thus
(3.23)
We also use the notation a2 for the scalar product of al' with itselU
Notice, however, that a2 need not be positive. Indeed, we can classifY all four-vectors
according to the sign of a2:
Ifa2 > 0,
Ifa2 <: 0,
If02
=
o,
al' is called timdike
al' is called spacdike
a" is called lightlike
(3.25)
From Ve!:tors it is a short step to tensors: a second-rank tensOr, sI'", carries two
indices, has 42
= 16 components, and transforms with two factors of A :
(3.26)
a third-rank tensor, tp";', has three indices, 4l = 64 components, and transforms
with thru factors of A:
(3.27)
• Th� closest thing is (a"b' - 0'''''), but this is
a =ond-rank Itnsor, not a four·vrctor (S<'e �
low).
t On its face, this is dangerously ambiguous
notation, sinc� "I could also bto the second
s,,",tial component of "-. But in practice we
so St"ldom ref.." to individual components
that this causes no problems (if you really
moan the compon�nt, btot"'r say SO explic·
itly). More serious is the potential confusion
btotwffn "I and � square of the magnitude
of the thrtt-vrctor pan of af'. [ p.."sonaUy
WI;'" the laruer in bold face, to aV(lid any po$'
sible misunderstanding: al a . a. This is
not standard notation, however, and if you
pr�fer some other device, that's fine. But !
do urge you to find a deu way to distinguish
"I from aI, or you are asking for real trouble
down the road.
_
1 95
961 ] R�IQlivisti, KilnmQli�
and so on. In this hierarchy, a vector is a tensor of rank one, and a scalar (invariant)
is a tensor of rank zero. We construct covariant and 'mixed' tensors by lowering
indices (at the cost of a minus sign for each spatial index), for example
(3.28)
and so on. Notice that the product of two tensors is itself a tensor: (a"b") is a tensor
of second rank; (a" t").,,) is a tensor of fourth rank; and so on. Finally, we can obtain
from any tensor ofrank 11 + 2 a 'contracted' tensor of rank n, by summing like upper
and lower indices. Thus �" is a scalar; t,,"" is a vector; a" tf'"l is a second·rank tensor,
3.3
Energy and Momentum
Suppose you're driving down the highway, and pretend for the sake of argument
that you're going at close to the speed of light. You might want to keep an eye
on two different 'times': if you're worried about making an appointment in San
Francisco, you should check the stationary clocks posted now and then along the
side of the road. But if you're wondering when would be an appropriate time to
stop for a bite to eat, it would be more sensible to look at the watch on your wrist.
For, according to relativity, the moving clock (in this case, your watch) is running
slow (relative to the 'stationary' docks on the ground), and so too is your heart
rate, your metabolism, your speech and thought, evtrything. Specifically, whi
l e the
'ground' time advances by an infinitesimal amount dr, your own (or proper) time
advances by the smaller amount dr:
d,
dr = ­
y
(3.29)
At normal driving speeds, ofcourse, y is so close to 1 that dt and dr are essentially
identical, but in elementary particle physics the distinction between laboratory
time (read off the clock on the waUl and particle time (as it would appear on the
particle's watch) is crucial. Although we can always get from one to the other, using
Equation 3.29, in practice it is usually most convenient to work with proper time,
because r is invariant - aU observers can read the particle's watch, and at any given
moment they must all agree on what it says, even though their own clocks may
differ from it and from one another.
When we speak of the 'velocity' of a particle (with respect to the laboratory), we
mean, of course, the distance it travels (measured in the lab frame) divided by the
time it takes (measured on the lab clock):
'"
v=d,
(3.30)
But in view of what has just been said, it is also useful to introduce the proper
vtlocity, I), which is the distance traveled (again, measured in the lab frame) divided
J.J EntIiY Qnd MomenfUm
by the proper time:'
(3.31)
According to Equation 3.29, the two velocities are related by a factor of y :
(3.32)
TJ = yv
i much easier to work with, for ifwe want to go from the lab system, S,
However, TJ s
to a moving system, S, both the numeratorand t� dmominator in Equation 3.30 must
be transformed - leading to the cumbersome velocity addition rule Equation 3.5 whereas in Equation 3.31 only tht numemtor transforms; dr, as we have seen, is
invariant. In fact, proper velocity s
i part ofa four·vector:
(3. 33)
whose zeroth component is
dIet)
d,f!
o
=
=
1) '" dr (lly) dt ye
(3.34)
(3.35)
Incidentally, 1),,1)" should be invariant, and it is:
(3.36)
They don't make 'em more invariant than that!
Classically, momentum is mass times velocity. We would like to carry this over in
relativity, but the question arises: which velocity should we use - ordinary velocity
or proper velocity? dassical considerations offer no clue, for the two are equal in the
nonrelativistic limit. In a sense, it's just a matter of definition, but there is a subtle
and compelling reason why ordinary velocity would be a bad choice, whereas proper
velocity is a good choice. The point is this; if we defined momentum as mv, then
the law of conservation of momentum would be inconsistent with the principle
of relativity (if it held in one inertial system, it would not hold in other inertia!
• Pro!",I vdocity is � hybrid quantity, n
i the
sense that distionte is measu� in the lDb
,
its velocity is zero. If my te,minology disrurbs
you. caU '1 the ·four·vdocity·. I .hould add
frame whereas time is measur� in the par·
th.t ..!though proper velocity is the more con·
tive ·proper· in this oontext. holding that this
locity is still the mo", "",ural qU.1ntity from
rick frame. Some people object to the adj�­
should � reserved for quantities measured
n the p,ntide frame. Of tourse. in its
enti",ly i
OW" frame the particle never moves at .11 -
venient quantity II"> ca!c..z.,t< with. ordinary ve·
the point of view of an observer watching a
particle fly past.
197
93 1
J R.lar;uillic I(i""matics
systems). But, if we define momentum as mTj, then conservation of momentum
is consistent with the principle of relativity (if it holds in one n
i ertial system, it
automatically holds in all inertial systems). I'll let you prove this for yourself in
Problem 3.12. Mind you, this doesn't guarantee that momentum is conserved ­
that's a matter for txptrilmnts to decide. But it does say that if we're hoping to
extend momentum conservation to the relativistic domain, we had better not define
momentum as mv, whereas mTj is perfectly acceptable.
That's a tricky argument, and ifyou didn't follow it, try reading that last paragraph
again. The upshot is that in relativity, momentum is defined as mass times proptr
velocity:
p :: mTJ
(3.37)
Since proper velocity is part of a four-vector, the same goes for momentum:
(3.38)
The spatial components of pi' constitute the (relativistic) momentum three·vector:
(3.39)
Meanwhile, the 'temporal' component is
(3.40)
For reasons that will appear in a moment, we define the relativisl«: energy, E, as
(3.41)
The zeroth component of PP, then, is Ejc. Thus, energy and momentum together
make up a four·vector - the energy-momentumfour.vector (Offour·molmntum)
(3.42)
Incidentally, from Equations 3.36 and 3.38 we have
(3.43)
which, again, s
i manifestly invariant.
The relativistic momentum (Equation 3.37) reduces to the classical expression
in the nonrelativistic regime (v « c), but the same cannot be said for relativistic
energy (Equation 3.41). To see how this quantity comes to be called 'energy: we
J.J EMrgy Dna Momel1lum
expand the radical in a Taylor series;
(3.44)
Notice that the second term here corresponds to the classical kinetic energy,
while the leading term (me2) is a constant. Now you may recall that in classical
mechanics only changes in energy are physically significant - you can add a
constant with impunity. In this sense, the relativistic formula is consistent with
the classical one, in the limit II « c where the higher terms in the expansion are
negligible. The constant term, which survives even when II = 0, is called the rest
energy;
(3.45)
the remainder, which is energy attributable to the motion of the particle, is the
relativistic kinetic energy;'
1
3 v·
T = m?(y - 1) = _mv2 + -m - + .
2
8
(;2
(JAG)
In classical mechanics, there is no such thing as a massless particle; its momen­
tum (mv) would be zero, its kinetic energy (imv1) would be zero, it could sustain
no force, since F = rna, and hence (by Newton's third law) it could not exert a
force on anything else - it would be a dynamical ghost. At first glance you might
suppose that the same would be true in relativity, but a careful inspection of the
formulas
(3.47)
reveals a loophole; when m = 0, the numerators are zero, but if II = c, the de·
nominators alse vanish, and these equations are indeterminate (0/0). So it is just
possible that we could allow m = 0, provilkd the panicle always travels at tm spud
of light. In this case, Equation 3.47 will not serve to define E and p; nevertheless,
Equation 3.43 still holds:
v = c,
E = Iplc
(for massless particles)
(3.48)
Personally, I would regard this 'argument' as a joke, were it not for the fact that
massless particles (photons) are known to exist in nature, they do travel at the speed
oflight, and their energy and momentum are related by Equation 3.48. So we have
, Notice t�t [ h�ve never menlioned ',eLotivis·
lie m.....' in �ll this. It i. a superfluous qwm·
Hty thaI .erve. no u.tful function. In CUt you
encounter it. the definition is "'"j _ Y"'; il
has died out because il differ.; from £ only
by a faclOr of ,2. Whatevtr can be uid about
"'"j could jusl as wdl be uid about £. For in·
stance, the '(on.erv�tion of reLttivistic mass'
is nothing but cons-ervation of tIIerg», with a
brIO. of cl divided out.
199
100
I3
R�!Qri�i'li, l(immGliCl
to take the loophole seriously. You may well ask: if Equation 3.47 doesn't define
p and E, what does determine the momentum and energy of a massless particle?
Not the mass (that's zero by assumption); not the speed (that's always c). How,
then, does a photon with an energy of 2 eV differ from a photon with an energy of
3 eV? Relativity offers no answer to this question, but curiously enough quantum
mechanics does, in the form of Planck's formula:
E = /Iv
(3.49)
It is thefrtquency of the photon that determines its energy and momentum: the
2-eV photon is red, and the 3-eV photon is pUIJ!k !
3.4
Collisions
So far, relativistic energy and momentum are nothing but tkfinitions; the physics
resides in the empirical fact that these quantities are conserved. In relativity, as in
classical mechanics, the cleanest application ofthe conservation laws is to collisions.
Imagine first a classical collision, in which object A hits object B (perhaps they are
both carts on an air table), producing objects C and D (Figure 3.2). Of COurse, C
and D might be the same as A and B; but we may as well allow that some paint
(or whatever) rubs off A onto B, so that the final masses are not the same as the
n
i the drama; if some wreckage, W, is left at the scene, then we would be talking
original ones. (We do assume, however, that A, B, C, and D are the only actors
about a more complicated process: A + B -"" C + D + W.) By its nature, a collision
is something that happens so fast that no emmal force, such as gravity, or friction
with the track, has an appreciable influence. Classically, mass and momentum are
always conserved in such a process; kinetic energy mayor may not be conserved.
3.4.1
Classical Collisions
1. Mass is conserved: mil. + mB = me + mD.
2. Momentum is conserved: p" + Ps = Pc + Po.
3. Kinetic energy may or may not be conserved.
Before
Fig. 3.2 A collision in which A + B ...... C + D.
After
3.4 Colliswns
I like to distinguish three types of collisions: 'sticky' ones, in which the kinetic
energy decreases (typically, it is converted into heat); 'explosive' ones, in which the
kinetic energy ilUreases (for e1Qmple, suppose A has a compressed spring on its
front bumper, and the catch s
i released in the course ofthe collision so that spring
energy is converted into kinetic energy); and elastic ones, in which the kinetic
energy is conserved,
(a) Sticky (kinetic energy decreases): T... + Ts > Tc + TD,
(b) Expl�ivt; (kinetic energy increases): T... + Ts < Tc + To,
(c) Elastic (kinetic energy conserved): T... + Ts = Tc + TD,
In the extreme case of type (a), the two particles stick together, and there is really
only one final object: A + B -+ C. In the extreme case of type (b), a single object
breaks in two: A -+ C + D (in the language of particle physics, A dtcays into
C + D),
3.4.2
Relativistic Collisions
In a relativistic collision, energy and momentum are always conserved, In other words,
all four components of the energy-momentum four-vector are conserved. As in
the classical case, kinetic energy may or may not be conserved.
1. Energy is conserved: E... + EB = Ec + ED.
2. Momentum is conserved: p... + ps = pc + PD.
3. Kinetic energy may or may not be conserved.
(The first two can be combined into a single expression: � + p'; = 1? + h..)
Again, we can classify collisions as sticky, explosive, or elastic, depending on
whether the kinetic energy decreases, increases, or remains the same. Since the
t.:ltal energy (rest plus kinetic) is always conserved, it follows that rest energy (and
hence also mass) increases in a stickycollision, decreases in an explosive collision,
and is unchanged in an elastic collision.
(a) Sticky (kinetic energy decreases); rest energy and mass
increase.
(b) Explosive (kinetic energy increases): rest energy and mass
decrease.
(e) Elastic (kinetic energy is conserved): rest energy and mass
are conserved.
Please note: exapl in elastic collisions, mass is not conserved.' For example, in the
decay lTD -+ Y + Y the initial mass was 135 MeV/,2, but the final mass is zero.
• In the old terminology. _ would �y that nlali';',,;, !!U.ss is con�rwd, but r<sl mass is not
1 101
'" I J R�lalivi'lic Kinema/ics
Here rest energy was converted into kimtic energy (or. in the absurd language of
the popular press, infuriating to anyone with the slightest respet:t for dimensional
consistency, 'mass was converted into energy'). Conversely, if mass is conserved.
then the collision was elastic. In elementary particle physics, there is only one
way this ever happens: the same particles come out as went in'- electron-proton
scattering (e + p ..... e + pI, for example.
In spite of a certain structural similarity between the classical and relativistic
analyses. there is a striking difference in the interpretation of n
i elastic collisions.
In the classical case, we say that energy is converted from kinetic form to some
'internal' fonn (thermal energy. spring energy, etc.), or vice versa. In the relativistic
analysis, we say that it goes from kinetic energy to rest energy or vice versa. How
can these possibly be consistent? After all, relativistic mechanics is supposed to
reduce to classical mechanics in the limit v « c. The answer is that aU 'internal'
forms of energy are reRected in the rest energy of an object. A hot potato weighs
more than a cold potato; a compressed spring weighs more than a relaxed spring.
On the macroscopic scale, rest energies are enormously greater than internal
energies, so these mass differences are utterly negligible in everyday life, and very
small even at the atomic level. Only in nuclear and particle physics are typical
internal energies comparable to typical rest energies. Nevertheless, in principle,
whenever you weigh an object, you are measuring not only the rest energies
(masses) of its constituent parts, but all of their kinetic and interaction energies
as well.
3.S
Examples and Applications
Solving problems in relativistic kinematics is as much an art as a science. Although
the physics involved is minimal - nothing but conservation of energy and conser·
vation of momentum - the algebm can be formidable. Whether a given problem
takes two lines or seven pages depends a lot on how skillful and experienced you
are at manipulating the tools and the tricks of the trade. I now propose to work a
few examples, pointing out as I go along some ofthe labor-saving devices that are
available to you {2J.
Example 3. J Two lumps ofday, each ofmass m, collide head-on at �c (Figure 3.3).
They stick together. QlUstion: What is the mass M ofthe final composite lump?
Solution: Conservation of energy says El + E2
EM. Conservation of momentum
says PI + P2 = PM. In this case, conservation of momentum is trivial: PI = -Pl, so
the final lump is at rest (which was obvious from the start). The initial energies are
=
up to th� sam� total mass, then I $UpPO� th� r�action A + B _ C + D might be considered
• In principle, iftMr� 61sted IW¢ distinct j»irs of j»rticles (A, B and C. D) that happen..d to add
'elastic'. but in reality there "", no such coincidences, $0 to a j»rticle physicist the word 'dastic'
has (orne to IrUaIl th.t the same j»rtides come OUI a$ went in.
3.5 fxamplu and Applioation<
------<>
m
00
M
After
Before
Fig. 3.3 Slicky collision of two equ�1 m�s'e� (Example 3.1).
equal. so conservation of energy yields
Mel = 2E", =
2..0
J1
5
(3/5)2
= -(2mc2)
4
Conclusion: M "" �m. Notice that this is grtallr than the sum ofthe initial masses;
in
sticky collisions kinetic energy is converted into rest energy. so the mass
increases.
Exomple 3.2
W-,
A particle of mass M. initially at rest. decays into two pieces. each of
mass m (Figure 3.4). QlUStion: What is the speed of each pie<:e as it Aies 0fI?
Solution: This is. of course. the reverse of the process in Example 3.1 Conservation
of momentum just says that the two lumps Ay off in opposite directions at equal
speeds. Conservation of energy requires that
so
J (2m/M)2
v=C
1-
This answer makes no sense unless M exceeds 2m: there has to be at least enough
rest energy available to cover the rest energies in the final state (any extra is fine;
it can be soaked up in the form of kinetic energy). We say that M
1m is the
threshold for the process M -+ 2m to occur. The deuteron. for example. is below
the threshold for decay into proton plus neutron (md = 1875.6 Mev/e2 ; mp + mK =
""
1877.9 MeVle2). and therefore is stable. A deuteron can be pulled apart. but only by
pumping enough energy into the system to make up the difference. (If it puzzles
you that a bound state of p and n should weigh k5s than the sum of its parts. the
point is that the binding energy of the deuteron - which. like all internal energy.
is reflected in its rest mass - is negative. Indeed. for any stable bound state the
bindng energy must be negative; if the composite particle weighs more than the
sum of its constituents. it will spontaneously disintegrate.) _
i
Exomple 3.3
A pion at rest decays into a muon plus a neutrino (Figure 3.S).
QutStion: What is the speed ofthe muon?
o
M
Before
,
-----0
,
0---m
m
After
fig. 3.4 A p�r1icle decays into two equal pieces. (Example 3.2).
1 10)
104 1 3
l?t1atj�isr.ic Kinematics
•
o
Before
Fig. 3.5 Df:cay of the charged pion (hample 3.1).
SoJutu,n: Conservation of energy requires E� = EI' + E Conservation of mo­
mentum gives p" = PI' + Po; but p" = 0, so PI' = -P Thus the muon and the
neutrino fly off back-to-back, with equal and opposite momenta. To proceed. we
nee<! a formula relating the energy of a particle to its momentum; Equation 3.43
does the job:
Suggestion 1. To get tlK energy of a particle, wlKn YOIl know its
momentum (or via versa), use the invariant
•.
•.
(3.50)
In the present case, then;
E,.. = m,,?
r
m
C
,
c
c
,,
+ ,"
�
,
EI' = cJ
Eo = Ip.I' = Ip"I'
Putting these into the equation for conservation ofenergy, we have
Solving for Ipl' I.
Meanwhile. the energy ofthe muon (from Equation 3.50) s
i
• You might be indin� to sol"" Equation 3.39 for the ""locity. and plu8 the result into
Equation 3.·41. but that would be a very poor strategy. Tn �neral. ",,]ocity is a bold parameter
to work with. in rdativity. Better to usc Equation 3.43, which tak� you dimily bold< and forth
between E and p.
3.5 Examples and Appljca/jans
Once we know the energy and momentum of a particle, it is easy to find its
velocity. If E = ymcl and p = y mv, dividing gives
pIE = vIc?
Suggestion 2. Ifyou know the energy and momentum of a JXir­
tick and you want to determine i/.5 velocity, use
(3.51)
So the answer to our problem is
=
Putting in the actual masses, I get til' 0.271c. DIll
There is nothing wrong with that calculation; it was a straightforward and
systematic exploitation of the conservation laws. But I want to show you now a
faster way to get the energy and momentum of the muon, by using four.vector
notation. (I should put a superscript /.I. on all the four-vectors, but I don't
want you to confuse the space-time index /.I. with the particle identifier /.I., so
here, and often in the future, I will suppress the space-time indices, and use
a dot to indicate the scalar product) Conservation of energy and momentum
requires
Taking the scalar product of each side with itself, we obtain
Bu'
Therefore
from which EI' follows immediately.
By the same token
Squaring yields
1 105
"' I J Re/alivisrk Kinemalics
which gives us IpI'l. In this case, the problem was simple enough that the savings
afforded by four-veclor notation are meager, but in more complicated problems the
benefits can be enormous.
Suggestion 3. Usefour-vuror nOlatien, and exp/Cjl the invari­
ant dot product. Remtmber that p� = m2c1 (Equation 3.43)
for any (nal) par1ide.
One reason why the use ofinvariants is so powerful in this business is thai we
are free 10 evaluate them in any inertial system we like. Frequently, the laboratory
frame is not the simplest one to work with. In a typical scattering experiment. for
instance, a beam of particles is fired at a stationary target. The reaction under study
might be, say, p + p-+ whatever, but in the laboratory the situation is asymmetrical.
since one proton is moving and the other is at rest. Kinematically, the process s
i
much simpler when viewed from a system in which the two protons approach
one another with equal speeds. We call this the ctnter-ofmomentum (eM) frame,
because in this system the total (three-vector) momentum is uro.
antiprotons, by the reaction p + P --l> P + P + P + p. That is, a high-energy proton
strikes a proton at rest, creating (in addition to the original particles) a proton­
antiproton pair. Question: What is the threshold energy for this reaction (i.e. the
Example 3.4
The Bevatron at Berkeley was built with the idea of producing
minimum energy of the incident proton)�
Solution: In the laboratory the process looks like Figure 3.&1; in the eM frame,
it looks like Figure J.6b. Now, what is the condition for threshold� Answer: Just
barely enough incident energy to create the two extra particles. In the lab frame, it
is hard to see how we would formulate this condition, but in the eM it is easy: all
four.finalparticles must be at rest, with nothing 'wasted' in the form ofkinetic energy.
(We can't have that in the lab frame, of course, since conservation of momentum
requires that there be some residual motion.)
hOT
Let
be the total energy-momentum four-vector in the lab; it is conserved,
so it doesn't matter whether we evaluate it before or after the collision. We'll do it
before:
�T =
(
E + me'
,
,
I P i , Q, Q
)
where E and p are the energy and momentum of the incident proton, and m is
the proton mass. Let
be the total energy-momentum four·vector in the eM.
Again, we can evaluate it before or after the collision; this time we'll do it aftu:
pt;.OT'
�' = (4mc, 0, 0, 0)
3.5 Exampl's amiApplirolions
(.(
0-­
P
p er'
P O--­
_ P o.....
P�
o
P
Af
ter
Before
'-0
1'(
j
/'t '\
Af er
Before
Fig. 3.6 P + P ..... P + P + P + p. (al In the lab frame: (b) in the eM frame.
since (at threshold) all four particles are at rest. Now Jl.;.OT :F P!;m', obviously, but
the invariant products PI'TOTJi;OT and P"ToTpft-OT' are equal:
Using the standard invariant (Equation 3.50) to eliminate pl, and solving for E, we
find
Evidently, the incident proton must carry a kinetic energy at least six limes its rest
energy. for this process to occur. (And n
i fact the first antiprotons were discovered
when the machine reached about 6000 MeV.) �
This is perhaps a good place to emphasize the distinction between a conserved
quantity and an invariant quantity. Energy is conserved - the same value aftu the
i not invariant. Mass is invariant - the same in all
collision as before - but it s
inertial systems - but it is not conserved. Some quantities are both invariant and
conserved (e.g. electric charge); many are neither (speed. for instance). As Example
3.4 indicates. the clever exploitation ofconserved and invariant quantities can save
you a lot of messy algebra. It also demonstrates that some problems are easier to
analyze in the eM system. whereas others may be simpler in the lab frame.
Suggestion 4. If a probkm sums cumbusoml in tht lab fmml.
try analyzing it in the eM system.
Even f
i you're dealing with something more complicated than a collision of two
identical particles. the eM (in which PTOT = 0) is still a useful reference frame. for
in this system conservation of momentum is trivial: zero before. zero after. But you
might wonder whether there is always a eM frame. In other words. given a swarm
and velocities VI. V2. Vl
does there
of particles with masses mI. ml. mj
necessarily exist an inertial system in which their total (three-vector) momentum
• . . . •
•
. •
1 107
"' 3 Reimivisfic Kinematics
I
l l prove it by finding the velocity of that frame and
is zero? The answer is yes; I wi
demonstrating that this velocity is less than c. The total energy and momentum in
the lab frame (S) are
ErOT = LYim;?;
proT = L YjmiVi
hOT
(3.52)
PrOT
is a {our-veclor, we can use the Lorentz transformations to get the
Since
momentum in system S'. moving in the direction of
with speed tI
In particular, this momentum is uro if v is chosen such that
t>
I L Yimov;l
C = EroT = L Y,miC
IPTOTlc
Now, the length ofthe sum ofthree·vectors cannot exceed the sum oftheir lengths
(this geometrically evident fact is known as the trial1glt inequality). so
and since Vi < C, we can be sure that v < c: Thus the eM system always exists.
and its velocity relative to the lab frame is given by
(3.S3)
It seems odd, looking back at the answer to Example 3.4, that it takes an incident
kinetic energy six times the proton rest energy to produce a pip pair. After all, we're
only creating 2mc2 of new rest energy. This example illustrates the inefficiency of
scattering offa stationary target; conservation ofmomentum forces you to waste a
lot of energy as kinetic energy in the final state. Suppose we could have fired the
two protons at one another, making the laboratory itself the eM system. Then it
would suffice to give each proton a kinetic energy of only mc2, one·sixth of what
the stationary·target experiment requires. This realization led, in the early 19705,
to the development of colliding-beam machines (see Figure 3.7). Today. virtually
every new machine in high.energy physics is a collider.
Example 3.5
Supposetwo identical particles, each with mass In and kinetic energy
T, collide head-on. QutStion: What is their relative kinetic energy, r (I.e. the kinetic
energy of one in the rest system of the other)?
• I am tacitly assumins that at \east on� of the
we
p.;orticks is massive. If aU of them an: =�s,
may obtain v : c. in which cas� th�r� is no CM system. For example, th�r� is no CM f....me
for a single photorL
3.5 Exampiesond ApplicGr.ions
A o--
--0 8
A 0>----
(b)
(.)
08
Fig. 3.7 Two e1<perimerltal arrangements: (a) Col liding �am5: (b) lixed target
Solutwn: 111ere are many ways to do this one. A quick method is to write down the
total four-momentum in the eM and in the lab
use Equation
3.50 to eliminate pi
and express the answer in terms of T = E - me2 and 1" = E' _ me2
(354)
lbe c/as5ical answer would have been 1"
= 4T, to which this reduces when T « mel.
(In the fE'st system of B, A has, classically, twice the velocity, and hence four times as
much kinetic energy. as in the eM.) Now, a factor of4 is some benefit, to be sure, but
the rdativistic gain can be greater by far. Colliding electrons with a laboratory kinetic
energyofl GeV, for example. would have a rtlativtkinetic energyof4000 GeV!
References
1 There are mtny excellent textbooks
on special relativity. I especially
r�ommffid Smith, J. H. (1996)
Introduction to Sp«iaJ Relativity.
Do,""r, N�w York. For � f�S(in�t·
ing {but unorthodox) approach, see
(�) nylor. E. F. and Wheeler. ,. A.
Problems
D
(1992) SpautJme PkjlSics. 2nd edn.
Freem�n, S�n F..mcisco. C.A.
2 [fyou w�nt to go into this
much mor� d��ply, th� Sl;m<.brd
r�f�r�n<:� is Hagedorn. R. (1964)
ReI(1tivistic Kin.malit;s. 6�njamin.
New York.
1.1 Solv� Equation 3.1 for x, y, Z, t in t�rm5 of X. y. ;t. t'. and check Wt you recov�r
Equation l.l.
1.2 (0) D�riv� Equation 3.4.
1 109
Rdafi�isfi, Kinematics
apart) �re both turned on al precisely g:oo P.M. Which one went on first according
(h) According to clocks on the ground (sySlem 5), SllHtlighlS A and B (situated 4 km
to an observer on a train (syslem 5'). which moves from A loward B at the speed
of Ught? How much later (in seconds) did the other light go on? Noll: As always in
�
relativity. we are \;lIking here about what 5' observed. after correcting for the lime it
took the light to reach her. not what she actually SI1'" (which would depend on where
she was located on the train).
,., (_j How do volumes transform? (lh container hn volume V' in its own rest frame. S.
whal is itsvolume as measured byan observer in S. with respect to which it is moving
at speed v?)
(hj How do dmsities transform? (If a con\;liner holds p' molecules per unit volume in its
own reSI frame. 5'. how many molecules per W1it volume does it orty in 5?)
H Cosmic ray muons are produced high in the atmosphere (al SOOOm. say) and travel
toward the earth at very nearly the speed of light (0.998 '. say).
6
I-I Given the lifetime of the muon (2.2 x 10- sec ). how rar would it go before
disintegrating, a«ording to prerelativistic physics? Would the muons make it to
ground level)
(bl Now answer the same question using rekltivisti, physics. (Because of time dilation.
the muons last longer. $0 they travel farther.)
(e) Pions are also produced in the upper atmosphere. In fact. the sequence is proton
(from outer space) hilS prolon (in atmosphere) ->p + p + pions. The pions then
decay into muons: 1f - -10 jJ.- + vu: 1f+ -> 1'-+ + �"" BUI. the lifetime of the pion is
much shorter (2.6 x 10-1 5). Assuming the pions have the same speed (0.998 f). will
they reach ground leveP
1.5 Half the muons in a monoenergelic beam decay in lhe first 600 m, How bst are they
going?
1.6 As the outlaws esope in their getaway or. which goes
Ie. the cop fires a bullet from
the pursuit car. which only goes Ie. The muzzle velocity (speed relative to goo) of the
bullet s
i ie. Does lhe bullet reach its target
1-) According to prerelativistic physics?
(h) According to relativity?
3.7 Find the matrix M that inverts Equation 3.12: xl' _ M�x"' (use Equation l.3). Show
that M is the matrix n
i verse of A: AM '" I.
l.3 Show thaI the quantity I (in Equation 3.13) is invariant under Lorentz transformations
(Equation 3.8).
3.9 Given two fouT·vectors, a" = (3. 4. 1. 2) and /II' _ (5, O. 3. 4], find: a
. P". al• t>l. a b.
"
al• pl. and a . b. Characterize al' and /II' as timdike. spacdike. or lightlike.
3.10 A second-rank lensor is c.alled symmtlric ifit is uO(:hanged when you switch the indices
(.'" -= 1'"); it is anlisymmelric ifit changes sign (a'" = -""").
la) How many independent elements are there in a symmetric tensor? (Since ,II "'" sl' .
these would count as only On� independent dement.)
(h) How many independentelements are there in an antisymmetric tensor?
(e) Show thaI symmetry is presenred by Lorentt transformations - thaI is. f
i 1" is
symmetric. so too is s""'. What about anlisymmetry?
3.5 &omp/e5 0l1dApplicl1tioll5
(dllf f'" is symmetric, show that$". is also symmetric. If a'" is antisymmetric, show
that a". is antisymmetTic.
(�l If 0'" is symmetric and a'" is antisymmt'tJic, show that s1'"a". '" o.
(I) Show t1utany second-ranktensor (t"') can bewritten as the sum ofan antisymmetnc
part (a"') and a symmetric p,art (s""): I'" = a'" + s"'. Construct 1'" and a'" explicitly,
given t''' .
3.!! A p;o.rticle is tr.Iveling at �' n
i the x direction. Detennine its proper velocity, rf (all four
components).
3.!2 Consider a collision in which p;utide A (with 4-momentum �) hits p<lrtide B (4_
momentum 1f.). producing p<lrticles C !i?) and D �). Assume the (relativistic) energy
and momentum are conserved in systems S (I,:. + p� = 1:: + to). Using the lorentz
transformations (Eq. 3.12), show that energy and momentum are also conserved in
S.
3.13 Is t' timelike, sp<lcelike, or lightlike for a (real) p;trtide of mass III? How about a
massless particle? How about a virtual particle?
3.U How much more does a hot potato weigh than a cold one (in kg)?
].15 A pion traveling at speed � decays into a muon and a neutrino, 11"- ->- /l- + iiI'. If the
neutrino emerges at 90° to the original pion direction, at what angle does the '"' come
off? IAllswer: tane .. (1 - III!/III!)/(2tlyl ).]
3.16 Particle A (energy E) hits particle B (at rest). producing particles C\. Cl. . . .:A + B ->- C)
+ Cl + . . . + C•. Calculate the threshold (i.e. minimum E) for this reaction, in terms
ofthe various particle masses.
[
Allswer :
3.17 Use the result ofProblem 3.1610 find the threshold energies for the following reactions,
assuming the target proton is stationary:'
(a)p + p ..... p + p + lfo
(b)p + p _ P + P + If+ + If(cl lf- + p _ p + p + n
(d)lf- + p _ K'l + 1:0
(eIP + p _ p + 1:+ + KO
].18 The first man-made n- (Fig. \.9) was created by firing a high�nergy proton at a
stationary hydrogen atom to produce a K+/K- p;o.ir: p + P _ P + P + K+ + K-;
the K- in tum hit another stationary proton, K- + P ..... n- + K'l + 1(+. What
minimum kinetic energy is required (for the incident proton), to make an n- in this
way? (GeU-Mann must have done this calculation to see whether the experiment would
be feasible.)
BewQ"': The Panick Ph)"ic5 Book/tl land
most other sources) list particle 'masses' in
MeV. For example, the mass of the muon
is quoted as 105.658 MeV. What they 1IItIlII,
ofCOUrsl'!. is the 1"<$1 <""'111' ofthe muon:
m"� = 105.658 MeV - or. what is the same
thing. m. = 105.658 MeVf�· !t is safest to
conven formulas from mass to rest mergy
bqOn plugging in any numbers. In this
case, for ex:omple, multiply top and bottom
by�, to get f.... '" UMel)l � (mAcl)1 �
Im1el)11I2Imlcl).
1 111
1121
J Re/atiwstic Kinematics
,.1, Particl� A. at r�t. decays into particl�s Band C (A ---> B + q.
(0) Fn
i d the energy ofth� outgoing particles. in terms ofthe various masses.
(h) Find w magnitudes ofth� outgoing momenta.
where .l.. is the so�alled trianglefoncticln :
.l..(:.:.)',z) ;;;; xl + r +r
- 2xy- 2:>.7- 2)'2'.
Ie) Note that A factors: .I.(al . Y. el) = (a + b + c)(a + II ')la - b + c)(a - h - c). Thus
IPDI goes to zero when m... mD + me. and runs imaginary if m" <: (mD + me).
=
-
Explain.
l.l0 U� the result of Problem 3.19 to find th� eM �nergy of each decay product in the
following reactions (see footnote to Problem 3.17):
I·I "- ...... /.I- +vp
(bIITQ ...... y + y
Ie) x:+ ...... IT'" + ,,0
IdI A ...... p + "'-
leI Q- ...... A + K-
1.21 A pion at rest decays into a muon and a neutrino (IT- ...... /.1- + vp). On th� average.
how far will the muon travel (in vacuum) befor� disn
i tegratingJ IAnswer: d = [(m.!
m!I/(2m.mu))cr = 1S6
m.1
1.22 Particle A. at rest. decays into three or more particles: A ...... B + C + D + .
-
('1 Determine the maximum and minimum energies that B can h�ve in such a decay. in
terms ofth� various mas�s.
(bl find th� maximum and minimum �k<:tron en�rgi� in muon decay. /.1- ...... e- +
v< + vU'
l.B (01 A p;ortide traveling at speed " approaches an identical particle at rest. What is
w speed (v) of each p<lrtide n
i w CM frame) (G:lssi<:lUl y. of course. it would
just be "/2,)
( - ;r=-urycr) 1
(Answer: (2'/11) 1
(h) find )' . 1/� in terms of1" • 1/-11 _ "l/el,
(Answer: -III" + 1)/2J
Ie] Use your result in p<lrl (b) to expr�s th� kinetic energy of �ach partid� in th� CM
frame. and thus re-derive Equation 3.54
l.U In reactions ofth� typt A + B ...... A + C, + C2 + . (in which p;orticl� A scatters off
particle B. producing Ct. Cl. . . -J. there is another inerti;tl frame, in addition to the lab
(B at rest) and the CM (PTOT = 0). which is sometimes u�ful. It is called the Brtit. or
'brick w;tll' frame. and it is the system in which A recoils with its momentum reversed
(pop.- _ -p�). as though it had bounced off a brid:: wall. Take the ca� of elastic
scattering (A + B ...... A + 8); if p;ortide A carri� energy E. and scatters at an angl� e,
in th� CM. what is its energy in th� Br�it frame? Find th� velocity of the Breit frame
(magnitude and direction) relative to the CM.
J.S Examples and Appliunions
US In a two-body sc;ottenng �nt. A + B .....
MandelslDm variables
5 l!!
C + D. it is con�nient to introd� the
{pJ, + PB)2/,2
;;;;; {pJ, - pc)2/,2
U = (PA - PDI2/C2
(a) Showtlut s + 1 + .. = m� + mi + m� + m�.
The Ik'QI"d;c,,1 virtue ofthe Mandelstom variables s
i that they are Lorentz invariants.
withthe same value in any inertialsystem. uP<"rimtl1IDUy.though.the moreaccessible
par:lmeters are energies and $(altering angles.
(b) Find the eM energy of A. in terms of s. I, u and the masses. {AI1nw:r: �M =
(s + m� - m�)2/2.fi.(
Ie) Find the Lab (B at rest) energy of A. (Answer: �b _ (s - m� - m�)cl/2m�_)
(d) Fn
i d thetotal CM energy (ETOT = EA + E� = Ee + ED). (Answer: EftA = .,rscl .)
3.U For elastic scattering of identica.! particles, A + A -> A + A, show that the Manddstom
variables (Problem 3.25) become
s = 4(p2 + m1(2)/2
=
_
u=
_
2p2(1 - cos O)/2
2p2(1 + cos 0)/2
where p is the eM momentum ofthe incident particle. and e is the scattering angle.
3.27 Work out the kinematics of Compton SGlttering: a photon of wa�length ), collides
elastiGillly with a charged panicle of mass m. If the photon scatters at angle e. find its
outgoing wavelength. ),'. (Answer: ),' "' ), +
(k{mc){l -cos 0).)
1 113
I ln
4
Symmetries
Symmdries play an important role in elementary particle physics, in part btcaust oftheir
relation kI conservation laws and in part because they permit Ont to make $Ome progress
when a complete dynamical theory is not yet available. The first section of this chapter
contains some general remarks about the mathematical description of�m1nttry (group
theory) and the rdation bttwten symmetry and conservation laws (Notther's theorem).
We then take up the case ofrotational �mmetry and its relation to angular momentum
and spin. This leads in turn to the 'internal' �mmetries - isospin, SU(J), and}lavor
SU(6). Finally, we consider 'discrete' symmetries - parity, charge conjugation, and
time reversal. Exctptfor the theory ofspin (Stclion 4.2) - which will be used txttnsilltly
in later chapters - and the material on parity in Section 4.1 - which is usefol
backgroundfor Chapttr 9 - this chapttr can be studied as supeljicially (or as deeply)
as the reader wishes. I recommend a quid: pass at this stage and a return to specific
i presupposed; readers
sections later, if warranted. Some knowledge of matrix theory s
fomiliar with quantum mechanie;s willfind Iht sections on angular momentum an easy
review (those unacquainltd with quantum mechanie;s mayfind them hoptltssly obscure,
in which case they should study the relevant chapttr ofan introductory quantum text).
Group theory is touched on here in a scandalously cursoryfashion (my main purpose is
to introduct some standard terminology); a serious student ofelementary particle physics
should plan eventually to study this subject infor greater detail.
4.1
Symmetries, Groups, and Conservation Laws
Take a look at the graph in Figure 4.1. I won't tell you what the functional form of
f(x) is, but this much is dear: It's an odd function, f(-x) = -f(x). (If you don't
believe me, trace the curve, rotate the tracing by 180°, and check that it perfectly
fits the originaL) From this it follows, for instance, that
ifl-xlr � iflxll'.
�
!I
-,.
j+>
j.,
l
�
_7
f(x) dx = 0,
[{(x)j2 dx = 2
+>
ior [{(x)f dx,
Inlroduaion I� &""'''lary POrlidu, StGotld EdiliolL O..id Griffiths
Copyright C 2008 WILEY·VCH VedasGmbH & Co. KGaA. Weinheim
ISBN, 97S-l·527-4060H
(4.1)
1161
.oj
5�mmtlrif5
Fig. 4.1 An odd function.
that no cosines appear in the Fourier expansion off(x), and that its Taylor series
contains only odd powers of x. In fact, you can deduce quite a lot aboutf(x), even
though you don't know its functional form, just from the observation that it has
a particular symmttry - oddness, in this case. In physics, intuition or a general
principle often suggests symmetries in a problem, and their systematic exploitation
can be an extremely powerful tool.'
The most striking examples of symmetry in physics are, [ suppose, crystals. But
we're not so much interested here in static symmetries of slw�as in dynamical sym·
metries of motion. The Greeks apparently believed that the symmetries of nature
should be directly reflected in the motion ofobjects: stars must move in circles be·
cause those are the most symmetrical trajectories. Ofcourse, planets do not, and that
was embarrassing (it was not the last time that naive n
i tuitions about symmetry ran
into trouble with experiment). Newton recognized that fundamental symmetries
are revealed not in the motions ofindividual objects, but in the set of(Ill possible mo·
tions - symmetries are manifest in the equations ofmotion rather than in particular
solutions to those equations. Newton's law of universal gravitation, for instance, ex·
hibits spherical symmetry (the force is the same in all directions), and yet planetary
orbits are elliptical. Thus the underlying symmetry ofthe system is only indirectly
revealed to us; indeed, you might wonder how we would ever have discovered it
from the observed planetary trajectories, if we didn't have a pretty strong hunch
that the gravitational field ofthe sun 'ought' to be spherically symmetrical.
It was not until 1917 that the dynamical implications of symmetry were com·
pletely understood. In that year, Emmy Noether published her famous theorem
• In som., r'"S�!$. th., ap!",al to symm�try
is ch..r..cteristic of an iJtcomplt� theory. for
example, f
i we som�how disc�red th� rx·
plicit form ofj(><). say.fl") = e-"> sinl")),
m.,n the theorems in Equ..tion U would lose
their lus!eT. Why bother with poo1* informa.
tion when we can have it QU? But even in a
m.1R1re theory. symmetry coIl5iderations of·
ten leod to dee!",! understanding and alcu·
lational simplification. For n
i stance. if you·re
called upon to integrate fl") from -3 to +3.
il �ys to notice thatj(,,) is odd. even if you
do know its functional form.
.oj.! Symmwits, GfOljps, and ConservOlion Laws
Table •.1 Symmetries �nd conservation laws.
Symmetry
Translation in time
Transbtion in sp,ace
Roution
Gauge transformation
Con5<!rvation law
�
�
�
�
Energy
Momentum
Angular momentum
Charge
relating symmetries and conservation laws:
Noether's Theorem: Symmetries +l' Conservation laws
Every symmetry ofnature yields a conservation law; conversely every conservation
law reflects an underlying symmetry. For example. the laws of physics are sym­
metrical with respect to tnms/ations in time (they work the same today as they did
yesterday) . Noether's theorem relates this invariance to conservation of energy. If
a system is invariant under translations in space, then momentum is conserved; if
.
it is symmetrical under rotations about a point, then angular momentum is con­
served. Similarly. the invariance of electrodynamics under gauge transformations
leads to conservation of charge (we call this an intt:mai symmetry. in contrast to
the space-time symmetries). I'm not going to provt Noether's theorem; the details
are not terribly enightening
l
[1]. The m
i portant thing s
i the profound and beautiful
idw that symmetries are associated with conservation laws (see Table 4.1).
I have been speaking rather casually about symmetries, and I cited some
i ely
examples; but what precs
is a symmetry? It s
i an operation you can perform
(at least conceptually) on a system that leaves it invariant - that carries it into a
configuration indistinguishable from the original one. In the case ofthe function in
Figure 4,1, changing the sign of the argument, x -0. -x, and multiplyi ng the whole
thing by -1. f(x) -0. -f{-x), is a symmetry operation. For a meatier example.
consider the equilateral triangle (Figure 4.2). [t is carried into itselfby a clockwise
rotation through 120Q (R+). and by a counterclockwise rotation through 120Q (R_).
by flipping it about the vertical axis a (R.). or around the axis through b (Rt.), or c
(Re). [s that all? Well, doing nothing (I) obviously leaves it invariant. so this too s
i a
symmetry operation, albeit a pretty trivial one. And then we could combine opera­
tions - for example, rotate clockwise through 2400_ But that's the same as rotating
countt:r clockwise by 1200 (I.e. R� = R_). As it turns out. we have already identified
all the distinct symmetry operations on the equilateral triangle (see Problem 4.1).
The set of all symmetry operations (on a particular system) has the following
properties:
1. Closurt: If Ri and Rj are in the set. then the product. RiRj ­
meaning: first perform Rj. then perfonn R;' - is also in the
set; that is, there exists some 14 such that R;Rj = 14.
• Note the 'backwards' ordering. Think of the r;ymmetty o�rations as acting on a system to thdr
right: RoRj[.o.) = R;iRj(.o.)j: Rj acts first. �nd then R; �cts on the result.
1 117
11S
I 4 Symmmin
,
'A
B
"-'-----!----'-"
c
•
Fig. 4.2 Symmetries of the e<:luil�ter�1 tri�ngle.
2. Identity: There is an element I such that lR; = R;I = R; for
all elements R;.
3. Inwrse; For every element R; there is an inwm, R;-I , such
that R;R;-
I = R;-I R; = I.
4. Associativity: Rj(RjRk) = (R,Rj)Rt.
These are the defining properties of a mathematical group. Indeed, group theory
need not commuU: R,Rj oF RjRi, in general. If all the elements do commute, the
may be regarded as the systematic study of symmetries. Note that group elements
group is called Abelian. Translations in space and time form Abelian groups;
rotations (in three dimensions) do no! [2J. Groups can be finite (like the triangle
group, which has just six elements) or infinite (for example, the set of integers, with
addition playing the role of group 'multiplication'). We shall encounter continucus
groups (such as the group ofall rotations in a plane), in which the elements depend
on one or more continuous parameters' (the angle of rotation, in this case), and
discrete groups, in which the elements may be labeled by an index that takes on
only integer values (aU finite groups are, of course, discrete).
As it turns out, most of the groups of interest in physics can be formulated as
groups of matrius. The Lorentz group, for instance, consists of the set of 4 x
4 A matrices introduced in Chapter 3. [n elementary particle physics, the most
common groups are of the type mathematicians call U(n): the collection of all
unitary n x n matrices (see Table 4.2). (A unitary matrix s
i one whose inverse
I = [I'.) [fwe restrict ourselves further to
is equal to its transpose conjugate: U-
unitary matrices with determinant 1, the group is called SU(n). (The 5 stands for
'
'special', which just means 'determinant 1 .) If we limit ourselves to real unitary
1 = 0.) Finally. the group of rea!.
matrices, the group is O(n). (0 stands for 'orthogonal'; an orthogonal matrix is
orthogonal, n x n matrices of determinant 1 is SO(n); 50(n) may be thought of
one whose inverse is equal to its transpose: 0-
as the group of all rotations n
i a space of n dimensions. Thus, SO(3) describes the
• If this dependen� ukes the form of an Qnalytic function. it is called a Lit. group. AU of the con·
tinuous groups one encounters in physics are Ue groups 13].
Table 4_2 important symmetry groups.
Group name
V(n)
SU(n)
O(n)
SO(n)
Dimension
Matrices in group
""
"'"
"'"
"'"
unitary ( fl· V '" I)
unitary,determinant!
orthogonal {DO", I}
orthogonal, determinant 1
rotational symmetry ofour world, a symmetry that is related by Noether's theorem
to the conservation of angular momentum. Indeed, the entire quantum theory of
identical in mathematical structure to S U(l), which is the most important internal
angular momentum is really closet group theory. It so happens that SO(3) is almost
symmetry in elementary particle physics. $0 the theory of angular momentum, to
which we turn next, will actually serve us twice.
One final thing. Every group G can be represented by a group of matrices: for
every group element a there is a corresponding matrix M�, and the correspon­
dence respa:ts group multiplication, in the sense that if ab = c, then M"Mb = Me.
A representation need not be 'faithful': there may be many distinct group ele·
ments represented by the same matrix. (Mathematically, the group of matrices is
homomorphic. but not na:essarily isomorphic, to G.) Indeed, there is a trivial case,
in which we represent every element by the 1 x 1 unit matrix (which is to say, the
number 1). If G is a group ofmatrices, such as SU(6) or 0(18), then it is (obviously)
a representation of ilstlf - we call it thefondamental representation. But there will,
example, 5U(l) has representations ofdimension 1 (the trivial one), 1 (the matrices
in general, be many other representations, by matrices of various dimensions. FOr
themselves), 3, 4, S, and in fact every positive integer. A major problem in group
theory is the characterization of all the representations of a given group.
Of course, you can always construct a new representation by combining two old
ones, thus
IMil) I
(zeros)
But we don't count this separately; when we list the representations of a group,
we are talking about the so-called irreducible representations. which cannot be
da:omposed into block-diagonal form. Actually, you have already encountered
several examples ofgroup representations, prohably without realizing it: an ordinary
scalar belongs to the one-dimensional representation of the rotation group, 50(3),
and a va:tor belongs to the three-dimensional representation; four-va:tors belong
to the four..dimensional representation of the Lorentz group; and the curious
geometrical arrangements of Gell-Mann's Eightfold Way correspond to irreducible
representations of the group 5 UP).
120
I S�mme!rie!
..
4.2
Angular Momentum
The earth, in its motion, carries two kinds of angular momentum: orbital angular
momentum, nnv, associated with its annual revolution around the sun, and spin
angular momentum, lw, associated with its daily rotation about the north-south
axis. The same goes for the electron in a hydrogen atom: it too carries both orbital
and spin angular momentum. In the macroscopic case, the distinction is not terribly
profound: after all, the spin angular momentum ofthe earth is nothing but the sum
total ofthe 'orbital' angular momenta ofall the rocks and dirt dods that make it up,
in their daily 'orbit' around the axis. In the case of the electron this interpretation
is not open to us: the electron, as far as we know, is a true point particle; its spin
angular momentum is not attributable to constituent parts revolving about an axis,
but is simply an intrinsic property ofthe particle itself(see Problem 4.8).
Classically, we are free to measure all three components of the orbital angular
momentum vector, L r mv, to any desired precision, and these components
can assume any values whatever. In quantum mechanics, however, it is impossible
in principle to measure all three components simultaneously; a measurement of
Lx, say, inevitably alters the value of 4, by an unpredictable amount. The best
we can do is to measure the magnitude of L, (or rather, its square: L2
L L)
together with one component (which we customarily take to be the z component,
= x
=
L,,). Furthermore, these measurements can only return certain 'allowed' values"
Specifically, a (competent) measurement of L2 always yields a number ofthe form
(4.2)
where 1 is a nonnegative integer:
1 = 0, 1,2,3,.
For a given value of I, a measurement of L,; always gives a result ofthe form
(4.3)
(4.4)
where m, s
i an n
i teger in the range -I to +1:
Inj = -1, - 1 + 1. . . . , -l , O, H , . . . , 1 - 1, l
(4.5)
(21 + 1) possibilities. Figure 4.3 may help you to visualize the situation. Here I 2,
so the magnitude of L is .J6f1. = 2.45 fl.; Lz can assume the values 2f1., Ii, 0, Ii, or
=
-
• J �m not going to p""" the quanti�tion lUles for �nguln momentum �nd irtllis m.1IteriaI is
new to you. t suggest that y<lU consult a textbook on quantum mecJu,nics. All J propose to do
.
here is summarize the essenti�l results _ will need in wh�t rollows.
4.2 Atlgu/arMQmenlum
,
"
o�----I
-h
Fig. 4.3 Possible orientations of the �ngul'lr momentum vector for I _ 2.
-21i,. Notice that the angular momentum vector cannot be oriented purely in the z
direction.
1111' same goes for spin angular momentum: a measurement of S2 S . S can
only returnvalues ofthe form
=
(4.6)
In the case of spin, however. the quantum number s can be a half-integer as well
as an integer:
(4.7)
For a given value of 5, a measurement of Sz must yield an answer of the form
(4.8)
where m, is an integer or half·integer (whichever 5 is) in the range -s to s:
m, = -s. -s + I. . . . , s - l,s
(4.9)
(25 + 1) possibilities.
Now, a given particle can be given any orbital angular momentum I you like, but
for each type of particle, the value of s is fix�d. Every pion or kaon, for example.
has 5 = 0; every electron, proton, neutron. and quark carries 5 = !; for the p, the
"', the photon, and the gluon, s = 1; for the t:.'s and the n-, s = �; and so on. We
call s the 'spin' of the particle. Particles with half.integer spin are jmnwns - all
1 121
1221 4 Symm�lmf
Table 4 3 Classification of particles by spin
Bo:rons (integer
Spin 0
-
Pseudoscalar mesoIl$
$pin)
Spin 1
Fermions (half·integer spin)
Spin !
Spin
Mediators
Quarks/Leptons
-
V�tor mesons
Baryon octet
Baryon d�uplet
�
+- Elementary
+- Composite
baryons, leptons, and quarks are fermions; particles with integer spin are bosoMs ­
all mesons and mediators are bosons (see Table 4.3).'
4.2.1
Addition ofAngular Momenta
Angular momentum states are represented by 'kets':
II ml ) or Is m, ) . lbus,
if I
say the electron in a hydrogen atom occupies the orbital state 13 -1) and the spin
state
I i 1), I am telling you that I = 3, ml = -1, S = l (which si unnecessary, of
course; ifit is an electron, s must be
!), and m. = t. Now, it may happen that we
are not interested in the spin and orbital angular momenta separately, but rather in
the total angular momentum, L + 5. (In the presence of coupling between L and 5
- tidal. if it's the earth-sun system; magnetic, for the electron-proton system - it
is the sum, and not Land 5 individually, that wi
l l be conserved.) Or perhaps we are
studying the two quarks that go to make a 1/1 meson; in this case, as we shall see,
the orbital angular momentum is zero, but we are confronted with the problem of
combining the two quark spins to get the total spin ofthe 1/1: 5 = 51 + 52. In either
case, the question arises: how do we add two angular monuntat
J = 11 + /2
(4.10)
Classically, of course, we just add the components. But in quantum mechanics
we do not have access to all three components; we are obliged to work with one
Ulml) and li2m2), what total angular momentum state(s) Um) do we get? The z
component and the magnitlUfe. So the question becomes: if we combine states
components still add, naturally, so
(4.11)
but the magnitudes do not; it all depends on the relative orientation of II and h
(Figure 4.4). Ifthey are parallel the magnitudes add, but if they are antiparallel the
• The terms 'fermion' and 'boson' refer to
the ru�s for constructing composite ....,e·
...
functions for identical p.ortides: boson
....""
... functions must "" symmelric un""r
intercmmge of any two particles, fermion
wl,,<,funClions "'" arni.ymmetric. This kads
to the Pauli exclusion principk for fermions,
and to pr<!found differences in the statis·
tiul mechanics of the two particle types.
The 'connection between spin and stati.tiu;'
(all fermions have �lf·;nteger spin and al!
bosons ha"" integer spin) is a deep theorem
in quantum field theory.
t t'U use the letter I for generic ansubr rno­
menrum _ it could "" orbital 1L), spin (S), or
some cornbine<l quantity.
4.2 Angular MomMtum
--- --
Fig. 4.4 Addition of �ngul�r moment�.
between these extremes. As it turns out. we get evayjfrom (h + ill down to
magnitudes subtract; in general, the magnitude of the ye.::tor sum is somewhere
in integer steps [41:
Ul -ill.
(4.12)
For instance, a particle of spin 1 in an orbital state I = 3 could have total angular
2
2
2
2
momentumj = 4 (i.e. J = 20/i. ). orj = 3 U = 1211 ), orj = 2 U2 =
6Iiz).
Example 4. 1
A quark and an antiquark are boundtogether, in a state ofzero orbital
angular momentum, to form a meson. QlUstion: What are the possible values of
the meson's spin?
Solution: Quarks (and therefore also antiquarks) carry spin !, so we can get
! + ! = 1 or ! - ! = O. The spin·O combination gives us the 'pseudoscalar'
The spin-l combination gives the 'Ye.::tor' mesons (p's, K*'s, IP, w) - 'vector' means
mesons (n's, K's, I), I)') - 'scalar' means spin 0, 'pseudo-' will be explained shortly.
spin 1.
1m
To add three angular momenta, we combine two of them first, using Equation
4.12, and then add on the third. Thus, if we allow the quarks in Example 4.1 an
orbital angular momentum I > 0, we get mesons with spin I + 1, I, and I
-L
spin (they are bosons). By the same token, all baryons (made up of three quarks)
Because the orbital quantum number has to be an integer, all mesons carry integer
must have half-integer spin (they are fermions).
Example 4.2
Suppose you combine Ihree quarks in a stale ofzero orbital angular
momentum. QUt:$tion: What are the possible spins of the resulting baryon?
Solution; From two quarks, each spin i we get a total angular momentum of
� + � = 1 or i - i = O. Adding in the third quark yields 1 + i = � or l - i = !
(when the first two add to I), and 0 + i = i (when the first two add 10 zero). Thus
Ihe baryon can have a spin of � or l (and the latter can be achieved in two different
ways). In practice, s = � is the decuplet, s = t is the octet, and evidently, the
quark model would allow for another family with s = i. (If we permit the quarks
to revolve around one another, throwing in some orbital angular momentum,
I
the number of possibilities increases accordingly - but the total will always be a
half-integer).
�
Well, Equation 4,12 tells us what total angular momenta j we can obtain by
combining jl and j2, but occasionally we require the explicit decomposition of
1 123
124 1 4 Symmetrin
2 x 1 /2
.,
W
I"
.,
--;;;·,n
,
->a
·,n
'n
312
,n
'312
-1/5
0/,
'12
.�
+1/2
.'12
-,n
,/0
,
.,n
'"
,
'"
'/0
,
-,
3JO
'12
'12
-2/5
-1/2
- ,n
31'
,/0
0/,
31'
,/0
-31'
-3/2
-,
-'n
-312
,�
-.�
.,n
-1/2
,
.,n
-
.�
-,
'/0
- ' 12
'I'
- '12
,
Fig, 4,5 Clebsch-Gordan coefficients for), .. 2, )1 = 1(A square root sign over each number is implied.)
�,ml ) �2 mZ) into specific states oftotal angular momentum, �m):
VI+h)
�lm'}�lm2) =
L d.,l;(;' '''l�m),
with m = m, + ml
(4.13)
j-vl-hl
The numbers d�l"'l are known as Cltbsch-Goraatl roejficienis. A bookon advance<!
quantum mechanics will explain how to calculate them. In practice, we normally
look them up in a table. (Ibere is one in the P(lrticlt Physics Booklet, and the case
j, 2, jz = ! is reproduced in Figure 4.5) The Clebsch-Gordan coefficients tell
you the probability of gettingjU + 11fiz. for any particular allowedj, if we measure
p on a system consisting of two angular momentum states �,ml) and �2m2 ): the
probability is the sqmirf ofthe corresponding Clebsch-Gordan coefficient.
:=
Example 4.3
The electron in a hydrogen atom occupies the orbital state 12 1)
and the spin state 1 � �). Question: Ifwe measure p, what values might we get, and
what is the probability of each?
Solution: The possible values of j are 1 + s = 2 + � = and I s = 2 � = �,
The z components add: m = -1 + � = -to We go to the Clebsch-Gordan table
(Figure4.5) labele<!2 x which indicates that we are combiningjl = 2 withj2
and look for the horizontal row, labeled - 1 , �; these are the values of m, and m2 .
-
!
!,
Reading off the two entries, we find 12 - 1)1�
� j,
�) =
-
-
.fl l� - �) - Ii I �
= �,
-
So the probability of getting j = is
and the probability of getting j = � is
Notice that the probabilities add to 1, as, ofcourse, they must. W
t)·
�_
Example 4.4 We know from Example 4.1 that two spin.! states combine to give
spin 1 and spin O. Probltm: Find the explicit Clebsch-Gordan de<:omposition for
these states.
Solution: Consulting the 1 x � table, we find
4.2 Angu/arMomenlum
l)ll l)
= 111)
11
IH)I! - 1) = (-jz) I I0) + (72)100)
I � - i)l� 1) = (-31)110) - (-31)100)
It - !lI! - � ) = 11 - 1)
(4. 14)
Thus the three spin 1 states are
111) = 11 �)11 �)
(4.15)
whereas the spin 0 state is
(4.16)
By the way. Equations (4.15) and (4.16) (an be read directly offthe Clebs(h-Gordan
table; the (Oeffidents workboth directions:
Um) = I:di,.,;l;!,l mzUl mdUzmz)
(4.17)
li.i2
This time we read down the (olumns, instead of along the rows. The spin-l
(Ombination is (ailed the 'triplet', fOf obvious reasons, and spin 0 is called the
'singlet'. For future referen(e. notke that the triplet is
under inter(hange
of the partides, 1 ..... 2, whereas the singlet is antisyrnmetrk (that is, it (hanges
sign). Incidentally, in a singlet state the spins are oppositely aligned (antiparallel);
however, it is not the (ase that in a triplet state the spins are ne(essarily parallel;
they are for = 1 and = -1, but not for = O. '!D
sym�tric
m
m
m
4.2.2
Spin �
The most important spin system is s = 1; the proton, neutron, electron, all quarks,
and all leptons carry spin �. Furthermore, once you understand the formalism for
5 = 1, any other case s
i a relatively simple matter to work out. So I will pause here
to develop the theory of spin 1 in some detail.
A partide with spin 1 can have m. = ('spin up'), or m, = -1 ('spin down').
Informally, we represent these two states by arrows: t and -1-. But a better notation
is afforded by two<omponent column vectors, or spinors:
�
()
"
I
In) = 0 '
(4.18)
1 125
lui 4 Symmetries
It s
i often said that a particle of spin ! can only exist in one or the other of these
two states. but that is quite false. The most general state ofa spin· t particle is the
linwr combination
(4.19)
where a and 13 are two complex numbers. It is true that a mea,5urtrrn:nl of Sz
can only return the value +!Ii or -!Ii, but the first outcome, say, does not prove
that the particle was in the state t prior to the measurement. In the general case
(Equation 4.19), lal2 is the probability that a measurement of Sz would yield the
value +tfi. and 11312 is the probability of getting - �Ii. Since these are the only
allowed results. it follows that
(4.20)
Apart from this 'normalization' condition, there is no a priori constraint on the
numbers a and 13.
Suppose now that we are to measure S" or Sy on a particle in the generic state
given by Equation 4.19 What results might we get. and what is the probability of
each? Symmetry dictates that the allowed values be ±lfi - after all. it's perfectly
arbitrary which direction we choose to call z in the first place. But determining
the probabilities is not so simple. To each component of S we associate a 2 x 2
matrix:'
(4.21)
The eigenvalues of .I)" are
aret
X± _
-
±�, and corresponding normalized eigenve<:tors
7>
(±:h)
(4.22)
,
• Again, the ,uri...u;"" ofthese matrices will be
found in any quantum-medunics lex!.. My
purpose here is to show you Mil' angular mo
mentum is bandied in �rtide pbysics, not to
x=(:J
elCplain wily it is done this way.
t A nonzero column matrix
·
s
i called an eige"vtCtor of a giVl'n " x " lIllI­
n
ix M if
for some number ). (the eige"va/",,). Notice
that any multiple of X is still an eigenvector.
with the s.ame eigenvalue.
(see Problem 4.15). An arbitrary spinor
of these eigenvectors:
�)
4.2 At!g�/drMomMlum
can be written as a linear combination
(4.23)
where
(4.24)
The probability that a measurement of 5" will yield the value iii is lal2; the
probability of getting -tli is Ib12. Evidently, lal1 + Ibl1 = 1 (see Problem 4.16).
The general procedure ofwhich this was a particular instance, is as follows:
1. Construct the matrix, A. representing the obselVable A in
question.
,
2. The allowed values of A are the eigenvalues of A.
3. Write the state of the system as a linear combination of
eigenve<:tors of A: the absolute square of the coefficient of
the ith eigenvector s
i the probability that a measurement of
A would yield the ith eigenvalue.
Exomple 4.5
Suppose we measure (5,,)2 on a particle in the state
�)
Question:
What values might we get, and what is the probability of each?
Solution. The matrix representing (5,,)2 s
i thesquareorthe matrixrepresenting S,,:
S'" �
.'
4
( )
'
0
0
(4.25)
1
Since
¥. Thus we would be certain to
¥ (probability 1). The same goes for S� and S�, so every spinor is an eigenstate
of $2 $; + $; + S�, with eigen alue J�' . This should come as no surprise in
evtry spinor is an eigenve<:tor of S!, with eigenvalue
get
=
v
general for spin s we must have 52 * + l)lil. U
For mathematical purposes, the factor of � in Equation 4.21 is ugly, and it is
customary to introduce the Pauli spiFl matrices:
,
_
""
(4.26)
1 127
128
I 4 Symmtlrits
that S (�)17 The Pauli matrices have many interesting properties, some of
which are explored in Problems 4.19 and 4.20. We shall encounter them repeatedly
in the course ofthis book.
In a sense, spinors (two-component objects) occupy an intermediate position
between scalars (one component) and vectors (three components). Now, when you
rotate your coordinate axes, the components of a vector change, in a prescribed
manner (see Problem 4.6), and we might inquire how the components of a spinor
transform, under the same circumstances. The answer [SI is provided by the
following rule:
SO
=
_
(4.27)
where U(8) is the 2 x 2 matrix
U(8) = e-i("�)/2
(4.28)
The vector 8 points along the axis of rotation, and its magnitude is the angle of
rotation, in the right.hand sense, about that axis. Notice that the exponent here is
itself a matrix! An expression of this form is to be interpreted as shorthand for the
power series:
(4.29)
(see Problem 4.21): As you can check for yourself (Problem 4.22), U(9) is a unitary
matrix of determinant 1; in fact, the set of all such rotation matrices constitutes
the group SU(2). Thus spin· t particles transform under rotations according to the
two-dimensional representation of S U(2). Similarly, particles of spin I, described by
vtctcrs, belong to the three-dimensional representation of SU(2); spin. � particles,
described by a four-component object, transform under the four·dimensional
representation of SU(2); and so on. (The construction ofthese higher-dimensional
representations is explored in Problem 4.23.) You're probably wondering what
SU(2) has to do with rotations; well. as I mentioned earlier, SU(2) is essentiallyt
the same group as SO(3), the group of rotations in three dimensions. Particles of
different spin, then. belong to different representations of the rotation group.
&W<11T For nutrices it is not th� case that
�A�a '" �u, in Ir'n�raL You might wan! to
meck this by using th� mltrices in Prob!�m
•.21. H�, tM usu.J rule dots lpply if A
lnd B commute (i.�. ifAB '" BA).
t Ther� s
i ldually l subtl� distinction �!W�n
SU(2) lnd S0(3). According to Problem •.21.
th� matrix U for rotation through In angle
of 21I is -1: a spinor <hang<:< sign under su�h
l rOlltion. And �. gco"",trical/y. l rOlltion
through br is «Juiv.J�nt to no rotation at lll.
•
SU(2) is a kind of 'doublt<!' ""rsion of sam,
in which you don'! com� b.ack !o th� begin·
ning until you hl"" turnt<! through 720'. In
this sense, spinor r�pr�s�nlltions of SU(2)
lrt not 'trut' r�pr�ntation5 of tht rotation
group, and that's why !h�y do not lppeU in
classical physic.s. In quantum mfihanics only
th� squa" of th� wa"" fun�tion carries phys·
ical si8nifi�anc�, and in the squaring th� mi­
nus sign goes away.
4.3 F/QlIOr Symmerries
4.3
Flavor Symmetries
There's an extraordinary thing about the neutron, which Heisenberg observed
shortly after its discovery in 1932: apart from the obvious fact that it carries
no charge, it is almost identical to the proton. In particular, their masses are
astonishingly dose, mp "" 938.28 MeV/c2 , m" "" 939.57 MeV/c2• Heisenberg [6]
proposed that we regard them as two 'stales' of a singlt particle, the nucleon.
Even the small difference in mass might be attributed to the fact that the proton
is charged, since the energy stored in its electric field contributes, according to
Einstein's formula E = mel, to its inertia. (Unfortunately. this argument suggests
thallhe proton should be the heavier ofthe two, which is not only untrue, but would
be disastrous for the stability of matter. More on this in a moment.) If we could
somehow 'turn off' all electric charge, the proton and neutron would, according to
Heisenberg, be indistinguishable. Or, to put it more prosaically, the strong forces
experienced by protons and neutrons are identical.
To implement Heisenberg's idea, we write the nucleon as a two-(omponent
column matrix
(4.30)
with
(4.31)
This is nothing but notatio n, of course, but it is notation seductively reminiscent
of the spinors we encountered in the theory of angular momentum. By direct
analogy with spin, S, we are led to introduce isospin, I.o However, I is not a vector
in ordinary space, with components along the coordinate directions x, y, and z, but
rather in an abstract 'isospin space', with components we will call 1l, 12, and Il. On
this understanding, we may borrow the entire apparatus of angular momentum,
as developed earlier in the chapter. The nucleon carries isospin
and the third
component, fl' has the eigenvaluest + l (the proton) and -l (the neutron):
!,
(4.32)
The proton s
i 'isospin up'; the neutron is 'isospin down'.
This is still just notation; the physics comes in Heisenberg's proposition that the
slrong interactions are invariant under rotations in isospin space, just as, for example,
electrical forces are invariant under rotations in ordinary configuration space. We
• The word derives from th� misle�ding older term i.otopic .pin (introduced by Wigner in 1937).
Nuclear physicists use the (beller) word Iscb<>ric spin.
t Th� is no factor of Ii in this case; isospin is dimensionkss.
1 129
no
I " Symmetries
call this an inttmal symmetry, because it has nothing to do with space and time,
but rather with the relations between different particles. A rotation through 180¢
about axis number 1 in isospin space converts protons into neutrons, and vice
versa. Ifthe strong force is invariant under rotations in isospin space, it follows, by
Noether's theorem, that isospin is CQnstrvtd in aU strong inttnution$, just as angular
momentum is conserved inprocesses with rotational invariance in ordinary space.'
In the language ofgroup theory, Heisenberg asserted that the strong interactions
are invariant under an internal symmetry group SU(2), and the nucleons belong to
the two-dimensional representation (isospin �). In 1932, this was a bold suggestion;
today the evidence is all around us, most conspicuously in the 'multiplet' structure
of the hadrons. Recall the Eightfold Way diagrams in Chapter 1: the horizontal
rows all display exactly the feature that caught Heisenberg's eye in the case of the
nucleons; they have very similar masses but different charges. To each of these
multiplets, we assign a particular isospin I, and to each member of the multiplet,
we assign a particular f). For the pions. I = 1:
(4.33)
for the A, 1 ",, 0:
A = 100)
(4.34)
for the 6's, 1 = t:
and so on. To determine the i50spin of a multiplet, just count the number of
particles it contains; since I) ranges from -T to +1, in integer steps, the number of
particles in the multiplet s
i 21 + 1:
multiplicity = 21 + 1
(4.36)
The third component ofisospin, 1), is related to the charge, Q, oftheparticle. We
assign the maximum value, Il = I, to the member ofthe multiplet with the highest
charge, and fill in the rest in order of decreasing Q. For the 'pre·1974' hadrons those composed of u, d, and s quarks only - the expicit
l relation between Q and /J
is the Gdl.Mlll1n-NishijimaformulCi:
(4.37)
• It is tempting to overstat� th� so-alled
'<:har� indepoi'ndence' of the strong forc�s
(th� flct that they ar� the same for protons
"" for neutrons). It dCl"s MI S2Y t....t you
will get the same result if you substitute an
i"dil'idual proton for a neutron. only if you
inmchan� all protons and neutrons. (For
�nmple. there exists a bound state of th�
proton and th� neutron - the deuteron - but
th�r� is "0 bound state of two protons or two
neutrollS.) Indeoed. any such assertion would
b.. illComl"'tibl� with th� Pauli exclusion
prindpl�. sinc� l proton and a n�Uf{on Cln hoe
in th� sam� qlJ2ntum state. but two neutrollS
(or two protons) annot.
4.3 FIOl'f)r Symmdriel
where A is the baryon number and S is the strangeness.' Originally, this equation
was a purely empirical observation, but in the context of the quark model it follows
simply from the isospin assignments for quarks: u and d form a 'doublet' (like the
proton and the neutron):
(4.38)
and all the other flavors carry isospin zerot (see Problems 4.25 and 4.26).
But classification s
i not all that isospin does for us. It also has important dy.
lIamical implications. For example, suppose we have two nucleons. From the rules
for addition of angular momenta we know that the combination gives a total
isospin of 1 or O. Specifically (using Example 4.4), we obtain a symmetric isotriplet:
[11)
:.
1 1 0) =
pp
(:7i) (Pll+ lip)
(4.39)
11 - 1) = 1111
and an antisymmetric isosinglet:
100) =
(7i) (P1l - lIp)
(4.40)
Experimentally, the neutron and proton form a single bound state, the deuteron (d);
there is no boundstate oftwoprotons or oftwo neutrons. Thus the deuteron mustbe
an isosinglet. Ifit were a triplet, all three states would have to occur, sincethey differ
only by a rotation in isospin space. Evidently, there is a strong attraction in the / = 0
channel, but not in the / = 1 channel. Presumably, the potential describing the
interaction between two nucleons contains a term of the form I(l) . 1(2), which takes
the value t in the triplet configuration and � in the singlet (see Problem 4.27).
Isospin invariance has implications, too, for nucleon-nucleon scattering. Con­
sider the processes
-
(a) p + p --> d + JT -t­
(b) P + II --> d + JTo
(4.41)
(c) 11 + 11 --> d + JT-
Since the deuteron carries / = 0, the isospin states on the right are [1 1),
[I 0), and 11-1), respectively, whereas those on the left arepp = I l l } , 1111 = 11 - I},
• Since Q, A. md S �te all conserved by the
dfi:tromagnetic forces. it follows thlOt I, s
i
also conserved. H�er, the other two com·
ponents (I, and Il). and lienee also I itself.
�re 1101 conserved in electromagnetic interac.
tions. For example. in the decay".o ..... Y +
y. I goes from I to O. As for the ",""ak intM.
actions. th� don't �n conse� S, so I, is
,
not conserved in �k processes (for aample
A .....
takes I, = O to IJ '"
t Since isospin pertains only to the strong
forc<:s, it is not a rdevant quantity for leptons.
For consis�ncy. aU leptons and mediators are
assigned isospin zero.
P +1I-
-!).
I lll
1321 4 SymmtU'its
pn = (*)(1 1 0) +
in
amplitudes
and
10 0»).' Only the T = 1 combination contributes (since the
each case is pure 1 = 1, and isospin is conserved), so the
final stilte
scattering
are in the ratio
(4.42)
As we shall see,t the cross
$lClwn,
thus
O"a : U. : U,
u, goes like the absolute square ofthe amplitude;
(4.43)
=2: 1:2
Process (c) would be hard to set up in the laboratory. but (a) and (b) have been
measured, and (when corrections are made for electromagnetic effects) they are
found to be in the predicted Tiltio {7J.
As a final example, let's consider pion-nucleon scattering,
TrN --+ TrN.
There are
six elastic processes:
Tr++p . .. Tr++P
(e) Tr-+p .... Jt-+p
(e)
+ n .... Tr°+n
--+
Tr° + P Tr° + P
(d) Tr++n .... Jt++n
if) 1f- +n .... Jt- +n
(b)
(a)
1f 0
(4.44)
and four charge-exchange processes:
Tr+ + n .... Tr°+p
(� lfo+n .... Jt- +p
= 1,
Tr°+p . .. Tr+ +n
(!) 1f- +p .... Tr°+n
(h)
(g)
Since the pion carries /
(4.45)
and the nucleon / = 1, the t01il1 isospin can be l or
So there are just two distinct amplitudes here: Jli1, for / =
�.
� , and Afl' for / = i.
From the Clebsch-Gorda.n tables we find the following decompositions:
Tr+ + p: Ill)]! l)
= I� J)
Tr° + p: 11 O)I� 1)
ftl� l) - (*) Ii 1)
Tr-+ p: 11 - IJl H) = (*) I� - 1 ) - fi l l - i)
Tr+ + n: Ill)]l - �) = (73) I� l) + /fl! !)
Tr° + n: 11 OJll - l) = /fl� - l) + (*) I� - l)
Tr-+ n: II - IJI! - !J = I� - �)
=
• Add Eq�tions •.3� and 4.•0
(4.46)
and the followins po,ragraph, I anlio!",'" !.oter r...ults, hut I ho� it � clear from the conl6!
how the Q.ku\ation proceeds. If you wish. skip these !wo paragraphs for now.
t The theory of sallerins amplitudes md cross sections will be devr\o� in Chapter 6. In this
Reactions (a) and (f) are pure f = �:
4.3 Flavor Symmelries
(4.47)
The others are all mixtures; for example,
(4.48)
(I'll let you work out the rest, see Problem 4.28). The (fOSS sections, then, stand in
the ratio
(4.49)
At a eM energy of 1232 MeV, there occurs a famous and dramatic bump in
pion-nucleon scattering, first discovered by Fermi tt Ill. in 1951 l81; here the pion
and nucleon join to form a short·lived 'resonance' state - the d. We know the d
carries 1 = so we expect that at this energy .Al » .AI, and hence
�,
U. : uc : Uj = 9 : 1 : 2
(4.50)
Experimentally, it is easier to measure the total CfOSS sections, so (c) and (j) are
combined:
utoI(lT - + p)
U,OI(IT+ + p)
=
3
(4.51)
As you can see in Figure 4.6, this prediction s
i well satisfied by the data.
In the late 1950s history repeated itself. Just as in 1932 the proton and neutron
were seen to form a pair, it was now increasingly clear that the nucleons, the A,
the r;'s, and the 8's together, constitute<! a natural grouping within the baryon
family. They all carry spin �, and their masses are similar. It is true that the latter
range from 940 MeV/cz, for Ihe nucleons, up 10 1320 MeV/c2, for the 8's, so it
would be stretching things a bit to argue thai they are all different states of one
particle, as Heisenberg had suggested for the proton and neutron. Nevertheless, it
was tempting to regard these eight baryons as a suptnnultiplet, and this presumably
meant that they belonged in the same representation of some enlarged symmetry
group, in which the SU(2) of isospin would be incorporated as a subgroup. The
critical question became: what is the larger group? (The 'Eight Baryon Problem',
as il was called, was not always phrased this way; at the lime. most physicists
were surprisingly ignorant of group theQry. Gell-Mann worked out most of the
formalism he needed from scratch, and only later learned that il was well known to
mathematicians.) The Eightfold Way was Gell-Mann's solution to the Eight Baryon
Problem. The symmetry group is SU(3); the octets constitute eight-dimensional
representations of SU(3), the decuplet a lO·dimensional representation, and so on.
Onething that made this casemoredifficult than Heisenberg's was that no naturally
1 133
l
l
<l
I Symmetries
<I
200
(1232)
".0
,"0
170
"
0
150
1<0
lJO
120
-< "Xl
"
.
,
0
110
90
eo
'"
(,
"
• •
I I
I \
I \
I
\
60
50
"
JO
20
10
I
I
I
I
,
\
,,- P
!/
(1688)
(1525)
\.
......
"
/
Il
/'
I \
/\ ....,
(1920)
,
\
I
-- -
(2190)
i
-=
�-""-=- ---�
--=-'"
0
900
1 100
1500
1700
\900
Mus of" p svstem (MeVIc1)
2100
2300
2500
Fig. <1.6 Tolal cross se(lions for " +p (solid lin�) and "-p
(dashed line) suttering. (50!lIU: Gasiorowkz, S. (l966)
ElemtnUlry PtJrlidt Physics, John Wiley & Sons, New York,
p. 29<1. R�printed by permission of John Wiley and Sons,
Inc.)
occurring particles fall into the fundamental (three-dimensional) representation of
SU(3), as the nucleons, and later the K's, the 3's, and so on, do for SU(2). This
role was reserved for the quarks: u, d, and s together fonn a three-dimensional
representation of SUP), which breaks down into an isodoublet (u, dJ and an
isosinglet (5) under S U(2).
Of course, when the charmed quark came along, the flavor symmetry group of
the strong interactions expanded once again - this time to SU(4) (some SU(4)
supermultiplets are shown in Figure L13}. But things barely paused there before
the arrival ofthe bottom quark, taking us to SU(S), and finally the top quark, SU(6).
4.3 F/(il'Of Symmtlrin
Table U Quark masses (MeV/el)
Quar1<fbvor
"
d
,
b
Bare man
2
5
9S
noo
"00
174000
Eff�iWl mass
336
3<0
<86
1550
4730
t77 000
W"ming:These numbers ne somewhat
s!"",ul..live and modd dependent [12).
However, there is an important caveat in this neat hierarchy: isospin, SU(2), is a
very 'good' symmetry; the members of an isospin multiplet differ in mass by at
most 2 or 3%, which is about the level at which ele<:tromagnetic corre<:tions would
be expected: But the Eightfold Way, SU(3), is a badly 'broken' symmetry; mass
splittings within the baryon octet are around 40%. The symmetry breaking is even
worse when we include charm; the At(udc) weighs more than twice the A(ud5),
although they are in the same SU(4) supermultiplet. It is worse still with bottom,
and absolutely terrible with top, which doesn't form bound states at all.
Why is isospin such a good symmetry, the Eightfold Way fair, and flavor SU(6)
so poor? The Standard Model blames it all on the quark masses. Now, the theory
of quark masses is a slippery business, given the fact that they are not accessible to
dire<:t experimental measurement. Various arguments [9] suggest that the u and d
quarks are intrinsically very light. about 10 times the mass ofthe ele<:tron. However,
within the confines of a hadron, their effective mass is much greater. The pre<:ise
value, in fact, depends on the context; it tends to be a little higher in baryons than
in mesons (more on this in Chapter 5). In somewhat the same way, the effe<:tive
inertia of a spoon is greater when you're stirring honey than when you're stirring
tea, and in either case it exceeds the true mass ofthe spoon. Generally speaking, the
effective mass ofa quark in a hadron is about 350 MeV1,2 greater than its bare mass
(10) (see Table 4.4). Compared to this, the quite different bart masses of up and
down quarks are practically irrelevant: they JUnction as though they had identical
masses. But the s quark is distinctly heavier, and the c, b. and t quarks are widely
separated. Apart from the differences in quark masses, the strong interactions treat
i ospin is a good symmetry be<:ause the effe<:live u and d
all flavors equally. Thus s
masses are so nearly equal (which is to say, on a more fundamental level, be<:ause
their bart masses are so small); the Eightfold Way is a fair symmetry be<:ause the
effe<:tive mass of the strange quark is not too far from that of the u and d. But
• Indeed. it used to be thought WI isospin
was an .:wet symmetry efthe streng interac·
liens. and oill of the symmetry breaking was
atlribut>ble to elenromagnelic contamination.
The fact that the "-p mass splittins s
i in the
wrong dirt'Ctien to be purely electromagnetic
WaS troubling. however, and we now ["'li""e
that SU(2) is only an �pprw;ima.1t symmetry
ef the mons interactions.
I US
1361 4 Symmttrits
the heavy quarks are so far apart that their flavor symmetry is severely broken. Of
course, this 'explanation' raises two further questions: (i) Why does the binding
of quarks into hadrons increase their effe<tive mass by about 350 MeV/c2? The
answer presumably lies within QCD. but the details are not fully understood [11].
(ii) Why do the bare quarks have the particular masses they do? Is there some
pattern here? To this question, the Standard Model offers no answer; the six bare
quark masses, and also the six lepton masses, are simply input parameters, for
now, and it is the business of theories beycnd the Standard Model to say where they
come from.
4.4
Discrete Symmetries
4.4.1
Parity
Prior to 1956, it was taken for granted that the laws ofphysics are ambidextrous; that
s
i , the mirror image ofany physical process also represents a perfectly possible phys­
ical process (13]. To be sure, we drive on the right (at least, Americans do) and our
hearts are on the left, but these are obviously historical or evolutionary accidents; it
could just as well have been the other way around. Indeed, most physicists held the
mirror symmetry (or 'parity invariance') ofthe laws ofnature to be self-evident. But
in 1956, Leeand Yang [14] were ledto wonder (for reasons we will come hack toat the
end ofthis section) whether there had been any txptriJ1UnmI test ofthis assumption.
Searching the literature, theywere surprisedto discover that althoughtherewas am­
ple evidence for parity invariance in strong and ele<tromagnetic processes, therewas
no confirmation n
i the case of weak interactions. They proposed a test, which was
carried out laterthat year by Wu (15]. to settle the s
i sue. In this famous experiment,
radioactive cohalt 60 nuclei were carefully aligned, so that their spins pointed in. say,
the z dire<tion (Figure 4.7). Cobalt 60 undergoes beta decay C-Oeo --+ 60Ni +t + v.),
and Wu re<orded the direction ofthe emitted electrons. What she found was that
most ofthem came out in the 'southerly' direction, opposiU to the nuclear spin.
That's all there was to it. But that simple obselVation had astonishing implica·
tions. For suppose we examine the mirror m
i age ofthat same process (Figure 4.8).
The image nucleus rotates in the opposite direction; its spin points downward. And
yet, the ele<trons (in the mirror) still came off downward. In the mirror, then, the
electrons are emitted preferentially in the same direction as the nuclear spin. Here,
then, s
i a physical process whose mirror image does net occur in nature; evidently
parity s
i not an invariance of the weak interactions. Ifit wtre, the electrons in Wu's
experiment wouldhave to comeout in equal numbers. 'north' and 'south', butthey
don't.
The overthrow of parity had a profound effect on physicists - devastating to
some, exhilarating to others \16]. The violation is not a small effect; as we shall
4.4 Dj"" eu
N
s
SymmWi"
s
\'
N
\'
Mirror
Fig. 4.7 In th� bet.;o dec�y of cob<llt 60. most Fig. 4.8 Mirror image of Figure 4.7: Most
electrons ire emitted in the direction oppo· ele<:lrons ire emitted parallel to the nucleu
site to the nuclear spin.
spin.
seE' in Chapter 9, it is in fact 'maximal'. Nor is it limited to beta decay in cohalt;
once you look for it, parity violation is practically the signature of the weak force.
It is most dramatically revealed in the behavior of the neutrino. In the theory
of angular momentum, the axis of quantization is. by convention, the z axis. or
course, the orientation ofthe z axis is completely upto us, but ifwe are dealing with
a particle traveling through the laboratory at velocity v. a natural choice suggests
itself: why not pick the direction of mctum as the z axis? The value of m,/s for
this axis is called the htJicity of the particle. Thus a particle of spin � can have a
helicity of +l(m, = �) or -lim, = -�); we call the former 'right-handed' and the
latter 'left-handed:- The difference is not terribly profound, however, because it is
not Lorentz·invariant. Suppose [ have a right.handed electron going to the right
(Figure 4.9a), and someone else looks at it from an inertial system traveling to the
right at a speed grtatu than v. From his perspe<tive, the electron is going to the left
(Figure 4.9b); but it is still spinning the same way, so this observer will say it's a
left-handed electron. In other words, you can convert a right.handed elet:tron into a
left-handed one simply by changing your frame of reference. 1bat's what I mean,
when [ say the distinction s
i not Lorentz-invariant.
Butwhat ifwe applied that same reasoning to a neutrino taken, for the moment,
to be massless, so it travels at the speed of light, and hence there is no observer
traveling faster? It is impossible to 'reverse the direction of motion' of a (massless)
neutrino by getting into a faster-moving reference system, and therefore the helicity
-
• In Chapter 9. I $I1all introduce a technic.al distinction between 'handedness' and MUdry, bllt for
th� moment [ will U� the t�rms interchangeably.
1 137
1381 .. Symmetries
8
,
,
•
•
,-
•
8
,
(b) Left·handed
IIIl Right·handed
Fig. ".9 Helicity. III (�) the spill �Ild velocity �,e p�r�llel (he·
!icity +1); til (b) they �'e antip�raHel (helitity - I )
.
of a neutrino (or any other massless particle)' is Lorentz·invariant - a fixed and
fundamental property, which is not an artifact of the observer's reference frame.
II becomes an important e"IKrilmntal matter to determine the helicity of a given
neutrino. Until the mid·fifties, everyone assumed that half of all neutrinos would
be left·handed, and half right·handed, just like photons. What they in fact discovered
was that
NEUTRINOS ARE LEFT·HANDED;
ANTINEUTRINOS ARE RIGHT·HANDED.
Of course, it's tough to measure the helicity of a neutrino directly; they're hard
enough to dete<:1 at all. There is, however, a relatively easy indire<:t method, using
the decay of the pion: n- -+ /-L- + vp. If the pion is at rest, the muon and the
antineutrino (ome out back to hack (Figure 4.10). Moreover, since the pion has spin
0, the muon and the antineutrino spins must be oppositely aligned.t Therefore, if
the antineutrino s
i right·handed, the muon must be right.handed too (in the pion
rest frame) - and this is pre<:isely what is found experimentally [17). Measurement
of the muon helicity, then, enables us to determine the antineutrino helicity. By the
same token, in n + de<:ay, the antimuon is always left·handed, and this indicates
that the neutrino is left·handed. By contrast, consider the decay of the neutral pion,
nO -+ )' + y. Once again, in any given decay the two photons must have the same
helicity. But this is an electromagnetic process. which respe<ts parity, and thus, on
the average, we get just as many right·handed photon pairs as left·handed pairs.
Not so for neutrinos; they only n
i teract weakly, and every one is left·handed; the
•
9
•
-
"
Fig. ".10 Deuy of rr - it rest.
•
•
;� .
8
•
• For massless particles. ollly the maximal value of 1",,1 occurs. For example, the photon (all ha""
"" = +1 or ..., : -I, but not m, '" O. So the heliciry of a massless I"'rticle s
; always ±L In the
case of the photon. these represent St40tes of left· and risht<ircular polari...tion. The a!Js<,nce of
m, = 0 corresponds to the absence oflongirudinal polarizatioll in classical optics.
t The orbital angular momentum (ifthere is any) points perpendkulu to the outgoing �locities,
50 il does not affttt this argument
4.4 Disc�te Symmetries
mirror m
i age of a neutrino does not exist: That is about the starkest violaion
t
of
mirror symmetry you could ask for}
In spite of its violation in weak processes, parity invariance remains a valid
symmetry of the strong and ele<tromagnetic interactions. It is useful, therefore, to
develop some formalism and terminology. First, a minor te<:hnical point: Instead
of rtjltetions, which oblige us to choose arbitrarily the plane of the 'mirror', we
will talk about inversions, in which every point is carrie<! through the origin to the
diametrically opposite location (Figure 4.11). Both transformations turn a right
hand into a left hand; in fact, an inversion is nothing but a refie<tion followed by
a rotation (1800 about the y axis, in the figure). Thus in the cases of interest (which
also possess rotational symmetry) it is a matter of indifference which one is used.
Let P denote inversion; we call it the 'parity operator'. If the system in question is
a right hand, P turns it into an upside-down and backward left hand (Figure 4.11b).
When applied to a ve<tor, a, P produces a ve<tor pointing in the opposite dire<tion:
Pta) = -a. How about the cross product of two vectors: c = a x b? Well, if P
changes the sign of a and of b, then c itself does net change sign: P(c) = Co Very
strange! Evidently, there are two kinds of vectors - 'ordinary' ones, which change
sign under the parity transfonnation, and this other type, of which the cross
product is the classic example, which do not. We call the former 'polar' Ve<:tors,
when the distinction must be drawn, and the latter 'pseudo' (or 'axial') ve<tors.
Notice that the cross product of a polar ve<tor with a pSlUM vector would be a polar
ve<tor.
You have encountered pseudovectors before, though probably without using this
language; angular momentum is one, and so is the magnetic field. In a theory
with parity invariance, you must never add a vector to a pseudovector. Consider,
for example, in the Lorentz force law: F
= q[E + (v x B)/e]; v is a vector, and B is a
pseudove<tor, so v x B is a vector, and it is legal to add it to E. But B itselfcould never
• This s
i too strong a statement. Th�.., (ould, T
supposr, I>r right·handed nrutrinos around,
but thry do not interact with ordin�ry mat·
ter by any mtchanism prrsrntly known. In
fact since we now know th�t neutrinos h�ve
a small bul nonuro mass. righl·handed nW'
trinos must Mist. Nonr of this, howrvrr, alters
the fact that when a ,,- decays, the emrrging
-
jJ
is right·handed in the eM frame and that
by itsrlf deslroys mirror symmetry.
By the way. �ck in 1929. shortly aftrr the
publication of Dirac's equation. Weyl pre·
srnte<! � ooutifully simple theory of mass·
less p;irtides of spin t. which had thr fearure
that they carried a
fixed 'handedness'. At the
time, Wey!'s thoeory aroused limited interest.
except for th� photon. which curies spin \.
since ther� wrre no massless particlrs known,
When Pauli introduced the nwtrino. in 1931,
you might supposr thlt h� would dust off
Wey!'s theory �nd put it to use. He did not.
Pauli rejtcted Wey!'s thoeory out of hand , On
thr ground that it violated mirror symmetry.
He lived to regret this mistake. and in 1957.
Weyj's theory was triumphantly vindicated.
t [I may occur to you, lS it did to many physi·
cists at the time, that if we simuhanoeously
convert �ll �rticles into their lntip;irti·
des. then a kind of mirror symmrtry is
restor..,]: the iml� of RjJ- + v"
__
R· ..... jJ+ + v� [with a Ieft·handed neutrino).
(with a right·handed antineutrinof becomes
which is prrfectly oby. This rrlh�tion w�s
some comfort. unti
l 1�. when it, too, was
shown to f�il. More on this in the following
se<:tions.
1 119
1<40
I 4 Symmelries
�r--�
�
f
----'�--.-_71"-----:'--,.-- - - -
/
I
,
�
�/
/
I
,
/
�'
x
y
(iI) Reflection lin the x-z plane)
(x, V,l) (X,-Y,l)
....
Ib) Inversion (.>t, y, z)
......
(-.>t, -y, -zl
Fig. 4.11 R�fI�t;ons and ;nv�rs;on$.
be addedto E. As we shall see, it is preciseiythe addition ofa vector to a pseudovector
in the theory ofweak interactions that leads to the breakdown of parity.
Finally, the det product oftwo polar vectors does not change sign under P, but
the dot product of a polar vector ilnd a pseudovector (or the triple product ofthree
vectors: a . (b x c)) dots change sign. So there are two kinds of scalars, too: the
'ordinary' kind, which don't change sign, and 'pseudoscalars,' which do. All this is
summarized in Tilble 4.S.'
If you apply the parity operator twice, of course, you're right back where you
started:
(4.52)
(The parity group, then, consists of just two elements: / and P.) [t follows that the
eigenvalues of P are ±1 (Problem 4.34). For example, scalars and pseudovectors
have eigenvalue + 1. whereas vectors and pseudoscalars have eigenvalue - l . The
hadrons ilre eigenstates of P and can be classified according to their eigenvalue
• � tuminology eru,nds very simply to spe.:i l relativity: "" = (�o. a) is called a pseudoV«tor if
a
....
its 5p<1lial components constitute � pseudovector PI�) "'�: p is a pseudO)SC>.iar if;1 g
nus itself under s!",WlI inversions P(P) = -po
into mi·
•
Table 4.5 S.:alars �nd vectors under parity
4.4 Di,U'�te Symmetrie,
s<:atar
P(s) = ,
PseudosaJar
P(p) = -P
P(v) = -v
Vector (or potu \l«tor)
Pseudovector (or ami \l«tor)
P(a) = a
justas they are classified by spin, charge, isospin, strangeness, and so on. According
to quantum field theory, the parity ofa fermion (half-integer spin) must be opposite
to that of the corresponding antiparticle, while the parity of a boson (integer
spin) is the same as its antiparticle. We take the qutlrks to have positivt intrinsic
parity, so the antiquarks are negative: The parity of a composite system in its
ground state is the product of the parities of its constituents (we say that parity
is a 'multiplicative' quantum number, in contrast to charge, strangeness, and so
on. which are 'additive').t Thus the baryon octet and decuplet have positive parity,
(+ l)l, whereas the pseudoscalar and vector meson nonelS have negative parity,
(-1)(+1). (The prefix 'pseudo' tells you the parity of the particles.) For an excited
state (of two particles) there is an extra factor of (_1)1, where J s
i the orbital
angular momentum [181. Thus, in general, the mesons carry a parity of (_1)1+1
(see Table 4.6). Meanwhile, the phoum is a IItc/or particle (it is represented by the
vector potential AJ.!): its spin s
i 1 and its intrinsic parity is -1.
The mirror symmetry of strong and electromagnetic interactions means that
parity is conserved in aU such processes. Originally, everyone took it for granted
that the same goes for the weak interactions as well. But a disturbing paradox arose
in the early fifties, known as the 'tau-theta puzzle'. Two strange mesons, called at
the time r and e, appeared to be identical in every respect - same mass, same spin
(zero), same charge, and so on - except that one of them decayed into two pions
and the other into three pions, states ofopposite parity:
(p = (_1)
(p = (_1)1
• This choice is completely arbitrary; we could
just as weU do it the other w�y around. In·
d�, in principle we could aS$ign positi�
parity to some quark fla"ors and negati� to
others. This would lead to a different set of
hadronic pMitiu, hut the """"lVOtion of p<lr·
ity would still hold. The rule stated here is ob·
viously the s;"'p/e3r. and it leads to the con·
ventioml assignments.
2
==
=:
+1)
(4.53)
-1)
t There is less to this distinction than
mttts the eye; in a sense. it results from
a notational anomaly. Scrupulous consistency
would require that we write the parity oper·
ator in uponential fonn. P = �x. with the
oper�tor K playing a Tole �n�logous to. say.
spin (Equation 4.28). The rigenv:l.lues of K
would be I) and corresponding to +1 and
-1 for P. �nd multiplication of parities would
correspond to addition of K.
1.
1 141
1.2 1 Symmelrin
4
Table •.6 Qu�ntum numb�rs of �om� meson nonels
spin
Orbital
N.
angubr momentum
,- ,
1=0
s=o
1 ", 1
s=o
1= 1
5=1
1= 1
Observed Non�1
K
J
0-'
,-,.0"
,..
,..
1 : =1
"
,
b,
.,
"
"
1=I
K
K,.
II;
K'
K,.
K',
1=0
�, �'
man (MeV/ell
Average
.00
"I. "I
900
>200
1'1./1
1300
,<00
., W
hio
hil
BOO
It seemed peculiar that two othelWise identical particles should carry different
really the sameparticll (now known as the J<:+), and parity is simply notconselVed in
parity. The alternative, suggested by Lee and Yang in 1956 was that r and B are
one ofthe decays. This idea prompted their search for evidence ofparity invariance
in the weak interactions and. when they found none. to their proposal for an
experimental test.
4.4.2
Charge Conjugation
Classical electrodynamics is invariant under a change in the sign of all electric
charges: the potentials and fields reverse their signs. but there is a compensating
charge factor in the lorentz law, so the forces still come out the same. In elementary
particle physics, we introduce an operation that generalizes this notion of'changing
the sign ofthe charge'. It is called
charge conjugation, C, and it converts each particle
into its antiparticle:
Clp) = IP)
(4.54)
'Charge conjugation' is something ofa misnomer, for C can be applied to a neutral
particle, such as the neutron (yielding an antineutron), and it changes the sign
of all the 'internal' quantum numbers - charge, baryon number, lepton number,
strangeness, charm, beauty, truth - while leaving mass, energy, momentum. and
spin untouched.
As with P, application of C twice brings us back to the original state:
(4.55)
and hence the eigenvalues of C are ±1. Unlike P, however, most of the particles
in nature are dearly
do"
not eigenstates of C. For if Ip) is an eigenstate of C, it follows
Clp) = ±Ip) = Ip)
(4.56)
4.4
Diunlt Symmtlrits
so Ip) and 11') differ at most by a sign, which means that they represent the
same physical state. Thus, only thcst particlts that art tJuir own antipartidts can be
eigtnstatts ofe. This leaves us the photon, as well as all those mesons that lie at the
center of their Eightfold.Waydiagrams: lfO, rJ, rJ', pO, 41. w, t, and so on. Because
the photon is the quantum ofthe electromagnetic field, which changes sign under
C, it makes sense that the photon's 'charge conjugation number' is -1. It can be
shown [19J that a system consisting of a spin.� particle and its antiparticle, in a
configuration with orbital angular momentum I and total spin s, constitutes an
eigenstate of C with eigenvalue (_1)1+1. According to the quark model, the mesons
in question are of precisely this form: for the pseudoscalars, ! 0 and
0, so
C = +1; for the vectors, I = 0 and s = 1, so C = -1. (Often, as in Table 4.6, C is
listed as though it were a valid quantum number for the entire supermultiplet; in
fact it pertains only to the central members.)
Charge conjugation is a multiplicative quantum number, and, like parity, it is
conserved n
i the strong and electromagnetic interactions. Thus, for example, the
lfO decays into two photons:
=
s=
(4.57)
(for n photons C = (-I)", so in this case C = +1 before and after), but it cannot
decay into three photons. Similarly, the w goes to lfO + y, but never to lfO + 2y. In
the strong interactions, charge conjugation invariance requires, for example, that
the energy distributions of the charged pions in the reaction
(4.58)
should (on average) be identical ]201. On the other hand, charge conjugation is
not a symmetry ofthe weak interactions: when applied to a neutrino (left·handed,
remember), C gives a left·handed antineutrino, which does not occur. So the
charge·conjugated version of any process involving neutrinos is not a possible
physical process. And purely hadronic weak interactions also show violations of C
as well as P.
Because so few particles are eigenstates of C, its dire<t application in elementary
particle physics is rather limited. Its power can be somewhat extended, ifwe confine
our attention to the strong interactions, by combining it with an appropriate
isospin transformation. Rotation by 1800 about the number 2 axis in isospin
space" will carry I) into -I), converting, for n
i stance, a rr + into a n - . If we
then apply the charge conjugation operator. we come back to If+. Thus the
charged pions are eigenstates of this combined operator, even though they are
not eigenstates of C alone. For some reason the product transformation is called
'G·parity':
(4.59)
•
Some �uthors use the num�r 1 axis. Obviously. any axis in the 1-2 pbne wi!! do the job
[ 143
14011 4 Symmdries
All mesons that carry no strangeness (or charm, beauty, or truth) are eigenstates of
G;· for a multiplet of isospin I the eigenvalue is given (Problem 4.36) by
where C is the charge conjugation number of the neutral member. For a single
pion, G = -1, and for a state with n pions
G = (-W
(4.61)
This is a very handy result, for it tells you how many pions can be emitted in
a particular decay. For example, the p mesons, with I = 1, C = -1, and hence
G
+1, can go to twe pions, but not to three, whereas the 1/>, the w, and the '" (all
I = 0) can go to three, but not to two.
==
4.4.3
CP
As we have seen, the weak interactions are not invariant under the parity transfor·
mation P; the cleanest evidence for this s
i the fact that the antimuon emitted in
pion de<:ay
(4.62)
always comes out left·handed. Nor are the weak interactions invariant under C, for
the charge·conjugated version of this reaction WQuld be
(4.63)
with a left-handed muon, whereas in fact themuon always comes out right-handed.
However, if we combine the two operations we're back in business: CP turns the
left-handed antimuon into a right-handed muon, which is exactly what we observe
in nature. Many people who had been shocked by the fall of parity were consoled
by this realization; perhaps, it was the combined operation that our intuition had
been talking about all along - maybe what we should have meant by the 'mirror
image' of a right.handed electron was a left·handed positron.t If we had defined
parity from the start to be what we now call CP, the trauma ofparity violation might
have been avoided (or at least postponed). It is too late to change the terminology
• K+. for elQrtlple. is ...." �n eigenst<lte of G. for � taus it to 1(1'. and C tak.. that to K!'. The
idu could lJ,:>, 6tendN to the K's, by using an appropr;"te SU(J) transfOrmation in pla� of Rl.
but since SU(3) is not .. very good symmetry of tlJ,:>, strong forces, there s
i tittle per�ntage in
doing 1'0.
f Incidentally. we could perfectly wen take electric charge to he a p�oscalar in classical dectro­
dynamics; E becom.. a pseudo�10r and B a »ector. but the results are ill the Same. It is rully
a matter of taste whether you say the mirror image of a plus charge is positive Or negative. But
i' seems simp!.., to say the charge does not change, md this is the stan<brd convention.
4.4 Discnu
5ymmtuits
now, but at least this helps to appease our visceral sense that the world 'ought' to
be left-right symmetric.
4.4.3.1
Neutral Kaons
CP invariance has bizarre implications for the neutral K mesons, as was first
pointed out in a classic paper by Gell-Mann and Pais (21]. They noted that the �,
with strangeness +1, can turn n
i to its antiparticle 7(!, strangeness -1
(4.64)
through a se.::ond-<lrder weak interaction we now represent by the 'box' diagrams
not � and 7(!, but rather some linear combination of the two. I n particular, we can
in Figure 4.12." As a result, the particles we normally obselVe in the laboratory are
form eigenstates of CP, as follows. Because the K's are pseudoscalars
(4.65)
On the other hand, from Equation 4.54
K'
K'
{
{
d
•
u
•
T
I
I
,
•
d
t
I
W
•
I
! w-
-
I
•
I
u
•
w-
----
u
,
T
,
•
d
•
,
u
w-
- - -- -
d
(4.66)
}.,
}.,
Fig. 4.12 Feynman diagrams contributing to �;=�. (There
are others, including those with one or both u quarks re·
placed by c or I.)
• The possibility ofsuch an inlerconversion i. almost unique to the neutral bon system; among
the ·stable' hadroru; the only other candida�. are rJlifll, Ifl(ff', and B'!1t: (problem 4.38).
1 145
1461 4 Symmdri�,
Accordingly,
(4.67)
and hence the (normalized) eigenstates of CP are
(4.68)
with
CPIKJl = IKJl
and
CPIKl) = -IKl)
(4.69)
Assuming CP is conservtd in the weak interactions, K, can only de<:ay into a
state with CP= +1, whereas K2 must go to a state with CP = -1. Typically,
neutral kaons de<:ay into two or three pions. But we have already seen that the
two.pion configuration carries a parity of+1, and the three-pion system has P = -1
(Equation 4.53); both have C = +1 Conclusion: Kl de<:ays into two pions; K2 de<:ays
into three pions (never two):'
(4.70)
Now, the 2:rr decay is much faster, be<:ause the energy released is greater. So if we
start with a beam of J<O's
the Kl component will quickly de<:ay away, and down the line we shall have a beam
of pure K2'S. Near the source, we should see a lot of21f events, but farther along
we expect only 3:rr decays.
Well . . . that's a lot to swallow. As Cronin put it, in a delightful memoir (22J:
SO these gentkmen, GeU-Mann and Pais, predicted that in ad·
ditlen to tM short·lived K mesons, tMre should be Iong.lived K
mesons. "T"hty did t
i �autijitlly, ekgantly and simply. 1 think
theirs is a papu orn: should read sometimejus/.for its pure
beauty of reasoning. It was published in 1M Physical Review in
1955. A very lovely thing! You get shivers up and down your
spine, tspecially when you find you understand it. AI the time,
many of tM most distinguished tMortticians thought this prt­
diction was reaUy baloney.
• Actu�lly. witlltlle rigllt combination oforbital �ngular momenta. it is possible to construct a
CP = +1 state of tile ,. ',._,.0 system, but wllile this might allow K, t(> �y (rarelyl into ],.,
it �s not alter tile critical fact tIl�t Kl cannot go to br.
4.4 Discnu Symmtlru,
But it wasn't baloney, and in 1956, Lederman and his collaborators discovered
the K2 meson at Brookhaven (23). Experimentally, the two lifetimes are
II = 0.895 X lO- IO sec
I2 "" 5.11 x lO-8 sec
(4.72)
so the K,'s are mostly gone after a few centimeters, whereas the K2'S can travel
many meters. Notice that K, and K2 are not antiparticles of one another, like �
and /(0; rather, each is its own antiparticle ( C "" -1 for Kl and C "" +1 for K2).
They differ ever·so·slightly in mass; experiments give [24)
(4.73)
The neutral kaon system adds a subtle twist to the old question, ·What is a
particle?' Kaons are typically produced by the strong interactions, in eigenstates of
strangeness (10' and /(0), but they duay by the wt'ak interactions, as eigenstates
of CP (K\ and K2). Which, then, is the 'real' particle? If we hold that a 'particle'
must have a unique lifetime, then the 'true' particles are Kl and K2: But we
need not be so dogmatic. In practice, it is sometimes more convenient to use
one set, and sometimes. the other. The situation is in many ways analogous
to polarized light. Unear polarization can be regarded as a superposition of
left·drcular polarization and right-circular polarization. If you imagine a medium
that preferentially absorbs right.circularly polarized light, and shine on it a linearly
polarized beam, it will become progressively more left-circularly polarized as it
passes through the material. just as a � beam turns into a K2 beam. But whether
you choose to analyze the process in terms of states oflinear or circular polarization
is largely a matter of taste.
4.4.3.2 CP Violation
The neutral kaons provide a perfect experimental system for testing CP invariance.
By using a long enough beam, we can produce an arbitrarily pure sample of the
long·lived species. Ifat this point, we observe a 2;r decay, we shall knowthat CP has
been violated. Such an experiment was reported by Cronin and Fitch in 1964.[25]
At the end of a beam 5 7 feet long, they counted 45 two pion events in a total of
22,700 decays. That's a tiny fraction (roughly 1 n
i 500), but unmistakable evidence
of CP violation. Evidently, the long-lived neutral kaon is not a perfect eigenstate of
CP after all, but contains a small admixture of K\;
(4.74)
•
This. incident�lly. w�s the position �dYOC�ted by GeIl·M�nn and his.
1 1"'7
1481 4
Symm�lri£s
The coefficient " is a measure of nature's departure from perfect
CP invariance;'
experimentally, its magnitude is about 2.24 x 10-3.
Although the effect is smail, CP violation poses a far deeper problem than parity
ever did. The nonconservation of parity was quickly incorporated into the theory
of weak interactions (in fact, part of the 'new' theory - Weyl's equation for the
neutrino - had been waiting in the wings for many years). Parity violation was
left·handed, not just 50.01% of them. Parity is, in this sense,
easier to handle precisely because it was such a
the weak interactions. By contrast,
large effect: aU neutrinos are
maximally violated, in
CP violation is a small effect by any measure.
Within the Standard Model, it can be accommodate..1 by including an empirical
phase factor IS) in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, provided that
there are (at least) three generations of quarks. Indeed, it was this realization that
led Kobayashi and Maskawa to propose a third generation ofquarks in 1973, before
even charm was discovered. [27]
The Fitch-Cronin experiment destroyed the last hope for any form of exact
mirror symmetry in nature. And subsequent study of the semileptonic decays of
t evidence of CP violation. Although 32% ofall KL'S
KL revealed even more dramaic
decay by the 3]1" mode we have discussed, 41% go to
(4.75)
Notice that
CP takes (a) into (b), so if CP were conserved, and KL were a pure
eigenstate, (a) and (b) would be equally probable. But experiments show [28] that
KL decays more often into a positron than into an electron, by a fractional amount
3.3 x 10-3. Here, for the first time, is a process that makes an absolute distinction
between matter and antimatter, and provides an unambiguous, convention·free
it is the charge carried by the kptonprt:ferentiaUyproduad
in the decay of the long.lived �utral K meson. The fact that CP violation permits
definition ofpositive charge:
unequal treatment ofparticles and antiparticles suggests that it may be responsible
for the dominance of matter over antimatter in the universe. [29] We will explore
this further in Chapter 12.
For almost 40years, the decayof KL was the only context in which CPviolationwas
observed in the laboratory. In 1981, Carter and Sanda pointed out that the violation
should also occur with the neutral B mesons. [30] To explore this possibility,
produce enormous numbers of FfJITfJ pairs [31J. By 2001, their detectors ('BaBar'
'B·factories' were constructed at SLAC and KEK (in Japan), designe..1 specifically to
and 'Belle', respectively) had recorded incontrovertible evidence of
in neutral B decays. [32Jt Unlike the kaon system, where
CP violation
CP violation is a tiny
effect in relatively common decays (such as Equation 4.75), for the B's it tends
to be a large effect in extremely rare decays. For example, the branching ratio
• This i. not tb� only rout� by which KL an decay to 21f; in the Standard Model. there is also �
snull 'direct' CP viol�tion th�t docs not inV'Ol� J:O ... 1(0 mixing. but i5 associated ins�ad with
th� so-calkd 'pntguin' diagram5 (Problem 4.4il). Dir«t CP viol�tion in KL .... 2>r W�5 confirmed
in 1999 [261.
t 'Dir«t' CP vioLotion in neulTlol B decays w�s confirmed by both l�hs in 2004 133).
4.4 Oj�rdt Symmdries
for If> -+ K+ + rr- is only 1.82 x 10-s , but this decay is 13% more common than
its CP 'mirror image' If _ K- + 1f+. So far, this is the only other system in which
CP violation has been detected.'
4.4.4
Time Reversal and the TCPTheorem
Suppose we made a movie of some physical process, say, an elastic collision of
two billiard balls. If we ran the movie backward, would it depict a possible physical
process, or would the viewer be able to say with certainty 'No, no, that's impossible;
the film must be running in reverse'? In the case of classical elastic collisions, the
'time·reversed' process is perfectly possible. (To be sure, if we put a
lot of billiard
balls in the picture, the backward version might be highly improb"bk: we would be
surpristd to see the balls gather themselves together into a perfect triangle, with a
single cue ball rolling away, and we would strongly suspect that the film had been
reversed. But that's just be<:ause we know it would be extraordinarily difficult to
set up the necessary starting conditions, such that all the balls would roll together
at just the right speeds and in just the right directions. Thus the initial conditions
may give us a clue to the 'arrow of time', but the laws governing the collisions
themselves work just as well forward as backward.) Until fairly recently, it was
taken for granted that all elementary particle interactions share this time-reversal
invariance. But with the downfall of parity, it was natural to wonder whether time
reversal was really so sacred.[36]
As it turns out, time reversal is a lot harder to test than P or C. In the first place,
whereas many particles are eigenstates of P, and some are eigenstates of C. none is
an eigenstate of T (the 'time-reversal operator', which runs the movie backward).t
So we cannot check the 'conservation of T' simply by multiplying numbers, the
(say. It + P _ d + y), and run it in reverse (d + Y _ It + pl. For corresponding
way we can for P and C. The most direct test would be to take a particular reaction
conditions of momentum, energy, and spin, the reaction rate should be the same
in either direction. (This is called the 'principle of detailed balance', and it follows
directly from time-reversal invariance.) Such tests work fine for the strong and elec­
tromagnetic interactions, and a variety ofprocesses have been checked. The results
have always been negative (no evidence of T violation), but this is hardly surprising.
•
1he� is some evidence for a?/a:' mixing (341.
and mor� w::ently r/'ff/l mixing PSI. but
as yet no evidence of CP violation in dth..
case. Bec.1Iuse the b quark - like the $ quark
- annot dec.1lY ,vithout crossing a genera·
tion boundary. the B mesons - liu the Ks tend to be �latively long·lived (10-11 s). The
, quark.. by contrast. ""n 80 to an J without
crossing a boundary. and that mak� the D
mesons short·lived (10-" s). This is one re..·
son the B system is a mOre promising place
to look for CP violation, ev.:n though LYs.",
e""i� to prodlKe.
t A particle an be identical to its mirror im·
age. and, f
i it's neutral. to its own antipar·
tide. but it ""n·t be identical to itself·going·
m.ckward·in·time (at least, not if anything
eYer happens to itl.
I l<I�
1 SO
I
-4
Symmetries
On the basis of our experience with P and C. we expect to see a failure of time
reversal in the weak interactions, if anywhere. Unfortunately, inverse-reaction
experiments are tough to do in the weak interactions. Take, for instance, the
typical weak decay A -0- p+ +JT-, The inverse reaction would be p+ + JT - -0- A,
but we are never going to see such a process, because the strong interaction ofthe
proton and the pion will totally swamp the feeble weak interaction. To avoid strong
and electromagnetic contamination, we might go to a neutrino process. But it is
notoriously difficult to do accurate measurements on neutrinos, and here we are
presumably looking for a very small effect. In practice, therefore, the critical tests of
Y invariance involve careful measurements of quantities that should be precisely
zero if Y is a perfect symmetry. The classic example is a static electric dipole
moment of an elementary particle.' Probably, the most sensitive experimental tests
are the upper limits on the electric dipole moments of the neutron (37] and the
electron:(38]
d" < (6 X 10-26 cm) e,
d. < (1.6 x 10-27 cm) e
(4,76)
where t is the charge ofthe proton; no experiment has shown direct evidence of Y
violation.
Nevertheless, there is a compelling reason to believe that time reversal cannot
be a perfect symmetry of nature. It comes from the so-called YCP theorem,
one of the deepest results of quantum field theory (39]. Based only on the most
general assumptions - Lorentz invariance, quantum mechanics, and the idea that
interactions are represented by fields - the YCP theorem states that the combined
operation of time reversal, charge conjugation, and parity (in any order) is an
exact symmetry of any interaction. It is simply impossible to construct a quantum
field theory in which the product YCP is not conserved, If, as the Fitch-Cronin
experiment demonstrated. CP is violated. there must be a compensating violation
of Y. Of course, like any assertion ofimpossibility, the YCP theorem may just be
a measure ofour lack of imagination; it must be tested in the laboratory, and that
is one reason it is so important to look for independent evidence of Y violation.
But the YCP theorem has other implications that are also subject to experimental
verification: f
i the theorem is correct, every particle must have precisely the same
mass and lifetime as its antiparticle.t Measurements have been made on a number
� mass
of particle-antiparticle pairs; the most sensitive test to date is the K?
-
• For an eI�m�n\.l.ry particle, the dipole m0-
ment d, would hav� to point along th� axis
ofth� spin, s; there is no other direction
available. But d is a vector, where,... s is a
f'S"udov«.ror, so a nonzero dipol� moment
would imply vio�tion of P. Similarly, s
changes sign unokr time reversal, but d dots
not, 50 a nonzero d would also (and more
int�=tingly) mean viobtion of T. For further
details, � Ramsey, ref. (32).
t This would alw fullow from C invarian�.
Ho�r, sin� we know that the latis signfi
i cant that the
eq�liry of nusses and lifetimes (also
magnetic moments, n
i cidentally, �l­
though th� have opposite signs) follows
from the far weaker assumption of TCP
symmdry.
ter is violatN, it
K? mass, is
4.4
Djscrtlt Symmetries
be less
difference, which, as a fraction of the
known to
than 10-18,
i relatively
So the TCP theorem is on extremely firm ground theoretically, and it s
secure experimentally. Indeed, as one prominent theorist has put it, if a departure
is ever found, 'all hell breaks loose'.
References
1 S�, for txamplt, Halun. F. and
Martin, A. D. (1984) QlUlm and
Ltpuml. John Wiley & Sons. Ntw
York. Section 14.2. For a mo�
complett (if som<:what d.1tw)
discussion: � (a) Hi!!. E. l.
(1951) Reviews ofMoIkm Physics,
23, 2S3.
2 If this fact is not familiar. you
might want 10 look at Halliday. D..
Rtsnick. R. and Walktr. J. (2001)
Fundamtntal! of Phyne., 6th <"dn,
John Wiley & Sons. N<:w York,
Section 11.3.
i For aC(essiblt trtatmtnts of group
th�ry � Tinkham. M. (2001)
GrOl<p Thorory ol1d Qual1h.m Me·
chanicl, Do�r, N<:w York; (a)
Lipkin. H. J. (2002) Lie GroUplfor
Pedestrians. Dover, Ntw York.
• �, for txample. Menbachtr. E.
(1998) Qual11um MecMl1jel, 3rd
edn. John Wiley & Sons. New
York, Chapter 17. Section S.
s S� Tinkham, M. (2001) Group
Thorory al1d Qua"'um M""all-
ics. Dover. N<:w York; (a) lipkin,
H, J. (2002) Lie Groupsfor Pedes·
IMI1l, Dover, N<:w York.90
Chapter 16, Sections 3 and 4.
G Heisenberg, W. (1932) Zeilschrijl Fur
P"l"ik, n, 1. An English trans·
lation of this classic p.a.per appears
in (a) Brink. D. M. (1965) Nude<>r
Forus, Pergamon. Elmsford, NY.
7 Fliagin, V. B. et al. (1959) So­
vie! Physin, jETP, 35 (S), S928 Anderson. H. l.. Fermi. E., Long,
E. A and Nagle, D. E. (1952)
Pkysiwl Rntiew. 85, 936.
9 De Rujub. A.. Georgi. M. and
Glashow. S. L. (1975) Physiwl Re­
vi""" 012, 147; (a) Weinberg, S.
(19n) Trallsaclions ofthe New Yorl:
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(1981) Ameriwll Journal ofP"ysics.
49, %2; (e) Gasser, J. and Ltutwyler,
H. (1982) Physics Reporl. 87. 78.
10 Holsttin. B. R. (1995) Ameri­
call Journal of P"ysics, 63, 14.
11 A qualitatively plausible mecha­
nism is suggested by the 'MIT Bag
Model'. Fret quarks of mass m. con·
fined within a spherical shell of ra·
dius R, are found to have an effec­
tive mass ....11" - Jm1+ {Iix/Rc)l,
where x is a dimensionless num­
ber �round 2.5. Using the radius
of the proton (say, 1.5 )( to-II cm)
for R, we obtain II1df = 330 MeV/el
for the up and down quarks. S�
Close. F. E. (1979) AI1 Illtroduc1;011 10 Quarks and Parlolls. Aca­
demic, London, Section 18.1.
II Bue masses are taken from the
ParlicSe Physics Bookltl (2006). The
light quark effective maSSts are
somtwhat lower in mtsons and
higher in baryons; the best·fit
values depend on the context.
S� Tables 5J, 5.5, and 5.6.
1} For an i!!umin�ting introduc­
tion to parity see Feynman. R. P..
Leighton, R. B. and Sands, M.
(1965) The Feyllmall L"'um 011
Pkysics, vol. Ill, Addison-Wesley,
Reading. MA. p. 17-22.
1( �, T, D. and Yang, C. N. (1956)
Physical Review. 104. 254.
IS Wu, C. S. et aL {19S7) PhYlical
Rel'i""" lOS, 1413. In the inter·
est of clarity I am ignoring the
formidable technical difficulties in­
volved in this experiment. To keep
the cobalt nuclei aligned, the sam­
ple had to be maintained at a tem­
perature of less than I' K for 10
minutes. Small wonder that no
earlier experiments had stumbled
on evidence of p.arity violation.
1 151
152
1
'I Symmetries
16 S�, for eu.mple, huli's letter to
Weisskopf in Pauli. W. (1964) C(>/­
I«ud Srnntijic P�pers. vol. I. (eds
R. Kronig and V. F. Weisskopf).
Wiley·lnt�ence. New yort.
p. xii: (a) Morrison. P. (April
1957) saentifo: American. 45.
17 Backenstoss. G. et aL (1961) Physical
Review LUlers. 6. 'lIS: (a) Bardon.
M. et al. (1961) Physical Review LeI·
ters. 7, 23. Earlier experiments an·
ticipated this result: (b) Goldha�r.
M.. Grod:6ns. L. and Snyder. A W.
(1958) Physical Review. 109. 1015.
18 This comes from the angular part of
the Sp<ltia[ wave function. YI'"(8.4').
See. for example, Tinkham. M.
(2003) Gro«p Theory and Qu�n·
lum Mechanics. Dover. New York;
(a) Lipkin. H. I. (2002) Lie Group,
for Pedes!rn.rI$. Dover. New York.
p. 186. (or Problem 5.3 below).
19 Perkins. O. H. (1982) lmroduaion
10 High·Enetgy Physics. 2 d edn.
Addison·Wesley. Reading. MA.
p. 99.
20 Baltay C. et al. (1965) Physi·
cal Review Lt.Uers. 15. 591.
U Gell·Mann. M. and Pais. A. (1955)
Physiml Review. 97. 1387. This paper
was written before the overthrow of
parity. but the essential idea remains
unchanged if we substitute CP for
their C. Of course. they didn't draw
a qunk diagram like Figure 4.12;
they based their argument for
Equation 4.&4 on the f;oct that both
It.
K" and j{) on dec.;ly into J'T+ +
J'T-. So K" � J'T+ +J'T- �
1.2 Cronin J. W. and Greenwood. M. S.
(July 1982) Physics TOIkIy, 38. Cronin
uses an unorthodox sign convention.
putting a -1 into our Equation 4.66.
but the physics is still the same.
D Lande. K. et al. (1956) Physiml Re­
204
view. 103, 1901.
26 A1avi.Harati. A. et al. (1999) Physical
Review Lelltrs, 83. 22: (a) Fanti. V. et
al. (1999) Physics Ldlers B, 465. HS.
Earlier claims (see (b) Schwarzschlld.
B. (October 1988) Physics Today. 17.
appear to have been premature.)
27 Kobayashi. M. and MasiGtwa. T.
(1973) Progress in Theorttical Physics,
49, 652.
28 Gjesdal S. et al. (1974) Physics Lt.!.
lers. 52B. 113: (a) Bennett. S. eta!'
(1967) Physical Review !.dlers. 19.
993: (b) Donan, D. et al. (1967)
PhysiC!11 Revitw Ldters, 19. 987.
29 Wilczek, F. (Oe<.ember 1980) S,ien­
![Ii<: Am.,.imn, 82.
30 Carter. A. B. and Sanda. A. I.
(1981) Physical Review D. 23. IS67.
For a comprehensive treatment
of CP violation S� (a)
Bigi,
I. I. and Sanda. A. I. (2000) CP
Viol�lion, Cambridge Univer·
sity Press. Cambridge, UK.
11 Quinn. H. R. and Witherell. M. S.
(October 1998) Sciel1lific American,
76.
II Abashian. A. et al. (2001)
Ph)l!i·
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Au�rt, B. et al. (2001) Physi·
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H Aubert. B. et al. (2004) Physi·
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14 Schneider. O. (2006) Rtvitw of Parti·
cle Physics. 836.
lS Aubert. B. et al. (2007) Physi·
cal Review Lt.tlers, 98. 21l801.
36 Excellent informal treatments
of time reversal are Overseth.
O. E. (October 1969) Scienlific
AnKrican, 88: and (a) Sachs.
R. G. (1972) S<;ie"",. 176. 587.
17 Harris. P. G. et al. (1999) Phys·
,ml Rtview !.ttters. 82. 904: (a)
Ramsey. N. F. (1982) Reports of
Progress in Physics. 45. 95.
The detection of so minute a
38 Regan. B. et al. (2002) Physiml
nating story. See. for example.
Wu. C. S. et al. (1957) Physiml Re·
39 The TCP theorem was discover.-d
mass difference is itself a fasci­
view, 105, 1413. Se.::t 16.13.1.
1.5 Christenson. J. H. et al. (1%04)
Physical Review l.etlers. 13, 138.
Rev'ew WIers. 88. 071805.
by Schwinger. J. and Luders. G. and
perfect.-d by Pauli. W. (1955) Niels
Bohr �nd It.e De"t/opmtnl of Physics,
(ed W. Pauli), McGraw·HiIl. New
4.4
York. At first no one p..id much ;o.t·
Diu"'l� Symmdries
individually. [t was only with the fan
of parity, ..nd especially with the fail·
tention to the TCP theorem, because
at that time everyone thought T. C.
ure of CPo that the importance of
and P we� all perfe.::t symmetlYs
this theorem was fully ..ppreciated.
Problems
IHil1t: One way to do this is to label the three (orners, is in Figure 4.2 A given symmetry
4.1 Prove that I, R.. R_. R." R�, and R.: a� cUI the symmetries of the equibteral triangle.
.
operation carries A into the position formerly occupied by A. B, or C. If A ..... A. then
either B ..... B and C ..... C. or else B ..... C and C ..... B. Take it from there.]
4.2 Construct a ·multiplication t;o.ble· for the triangle group, filling in the blanks on the
following diagram:
[
R.
R_
..
R.
..
[
R.
R_
..
..
R,
[n row i. columnj. put the product IqRj. Is this an Abelian group? How can you tell,
just by looking at the multiplication table?
u [aJ Construct ..
(�)
3 X 3 representation of the triangle group as follows: let D(R) be the
matrix representing operation R. lt acts on the column matrix
new column matrix
(�) (�),
= D(R)
to produce ..
where A' is the vertex now occupying the
location origin;illy held by A. Thus, for eltilmple.
o
o
Find the oth"'r five matrices. (You might want to chttk that multiplication or your
matrices fits the table you constructed in Problem (4.2).)
[bJThetriangle group, lik", any other group, has a trivial one·dim",nsional repr",s",ntation.
It alro has a l1ol1trivial, one-dimension.al representation, in which the elements are
net all represented by 1. Work out this second one-dimensional representation. That
is. figure out what nwnber (1 x I matrix) each group ..,tem",nt is r"'presffited by. Is
this represffitation faithful?
u Work out the symmetry group of .. squ;o.�. How many elem",nts does it havt:? Construct
the multiplication table, and detennine whether Or not the group is Abelian.
4.5 [-I Show thaI the set orall unitary 11 x 11 matrices constitutes a group. (To proy", dosure,
for instance, you must show that th", product oftwo unitary matrices is itselfunitary )
(bl Show that the set ofall n X 11 unitary matric",! with detenninant 1 constitutes a group.
Ie) Show that 0(11) is a group.
Idl Show that SO(I1) is a group.
.
1 153
15�
I
.(
Symmetries
Cartesian axes x, y, ue (a., <lr). What are its components (a�, a:) in a system X, y which
is rotated, counterclockwise, by an angle e, with respect to x, y? Express your answer in
the form ofa 2 x 2 matrix R(e):
•.6 Consider a vector A n
i two dimensions. Suppose its components with respect to
Show that R is an orthogonal matrix. What is its detenninant? The �t of rill such
rotations constitutes a group; what is the name of this group? By multiplying the
matrices, show that R{8 L )R(1I1) = R{8L + Ill): is this an Abelian group?
.., Consider the matrix
(� i).
_
[s it in the group 0(2)? How about SO(2)? What is its
effect on the vector A ofProblem (4.6)? Does it describe a possible rotation ofthe plane?
u Suppose we interprct the electron literlLUy as a classical solid sphere ofradius r. mass m,
tfi. What is the speed, u, ofa point on its 'equator'?
Experimentally, it is known that r is less than 10-Lu cm. What is the corresponding
eql.Ll.torial speed? What do you conclude from this?
u When you are adding angular momenta, using Eql.Ll.tion 4.12, it is useful to check your
results by counting the number of states before and after the addition. For instance. in
Example 4.1 we had twoql.Ll.rks to begin with, each could have m, = +t or m, = � so
there were four possibilities in all. Afiu adding the spins, we had onecombination with
spin 1 (hence m, = 1, O. or -1) and one with spin 0 (m, = 0) - again, four states in all.
spinning with angular momentum
-
.
(a) Apply this check to Example 4.2
(b) Add angular momenta 2, 1, and
i. list the possible values of the total angulu
momentum, and check your answer by counting states.
•.10 Show that the 'original' beta-decay reaction 11 .... P + t would violate conservation of
angular momentwn (all tlm�e particles have spin i). Ifyou were Pauli, proposing that
the reaction is really n ..... p + t + ii,. what spin would you assign to the neutrino?
4.11 In the decay t;++ .... P + Jf+, what are the possible values ofthe (CM) orbital angular
momentum quantum number, [, n
i the final state?
•.12 An electron n
i a hydrogen atom is in a state with orbital angular momentum quantum
number [ = 1. If the I<ll<l[ angulu momentwn quantwn number j is �, and the z
component of total anguln momentwn is tfi, what is the probability of finding the
electron with m, _ + � ?
4.\3 Suppose you had two particles ofspin 2, each in a state with 5, = O. [f you measured
the Ioial angular momentwn of this system. given that the orbital angular momentum
is zero, what values might you get, and what is the probability ofeach? Check that they
add up to 1.
u( Suppose you had a particle of spin f, and another ofspin 2. If you knew that their
orbital angular momentum was zero, and that the totlll spin of the composite system
was �. and its z component was -f. what values might you get for a measurement of
S� on the spin·2 particle? What is the probability ofeach? Check that they add up to \.
4.15 Check that x±. Equation 4.22, are nonnalized eigenvectors of �x, Equation •.21, and
find the associated eigenvalues.
4.16 Show that (,,(1 + Ih(l = 1 (Equation 4.24). provided the spinor in question is nonnalized
(Equation 4.20).
-4.17 (a) Find the eigenvalues and normalized eigenspinoTS of�r (Equation �.21).
(b) Ifyou measured Sr on all electron in the state
(.IS Suppose an el�ron is in the state
what is the probability of each?
(�).
(;) .
what values might you ge� and
4.4 DiscreTe SymmeTries
lal [fyou measured S., wh�tv�lues might you get, �nd wm.t is the prob.ability of each?
[hI [fyou measured Sr' what values might you get. and what is the probability of each?
Ie) If you menured S., what values might you get, and what is the prohability of each?
4.19 lal Show that o} _ 0:
o} _ 1. ('1' here really means the Z x 2 unit matrix; if no
matrix is specified, the unit matrix is understood.)
(bl Show that 17.0, = -uru. w.,of'"z - -17.0, = ia.,ozox = -0,0. = W,. These reo
sults are neatly summarized in the formula
..
"
'"
)
(summation over k implied), where dij is the Kronecker deltil:
$� =
I
0,
if ; :j
othelWise
and t(il is the Levi-Civita symbol:
"ijk =
! 1.
-I,
0,
ifijk. = 123,231,or312
ifijk. = 132,213,or 321
otherwise
4.2D Use the results ofProblem 4.19 to show that
..
(al The commutilt()I, [A, 8] • AB - BA, oftwo Pauli m�trices is [u;.O"jl
[h)The IIl'1ticcmmuUllOr. (A, BI _ AB + BA. is (0;, ujl
28q.
(() For any two vectors a and b, (a . a)(a . b)=a b + itT . (a x b).
4.21 (a) Show that d'<�,!l = ia,.
,.,
2��j"'�.
[hI Find the matrix U representing � rotation by 180" about the y axis, and show that it
conY<':rI$ 'spin up' into 'spin down', as we would expect.
(e) More generally, show that
• , A
sln
(I
U(B) = cos(I - 1(tt·cr)
2
2
where U(II) is given by Equatioo 4.28, e is the m�gnitude of ,I, and � . lI/e. IHinl:
Use Problem 4.20. part (e).1
4.ll (a) Show that U, in Equation 4.28, is ul'liUlry.
(b) Show that det U = 1. (Hint: You can either do this directly (however, see footnote
after Equation 4.29). or else use the results ofProblem 4.21.1
4,l] The utension ofeverything in Section 4.2.2 to higher spin is relatively straightforward.
For spin 1 we have three states (m, = +1.0, -I), which we may represent by column
vectors:
respectively. The only problem is \0 eonstroct the 3 x 3 matrices S
latter is easy:
•.
S, and S The
•.
I 15S
156
I 4 Symmtld,s
lal Construct S, for spin L To obtain S", and Sr it is easiest to start with the 'raising' and
'lowering' operators, Sot lIIS", ± i�, which han the property
..,!s (s+ 1)
S"lsm) '= Ii
m(m
± l)1s(m ± 1))
(4.77)
(1)1 Construct the matrices S+ and S_, for spin l.
(el Using (b). determine the spin·I matrices S� and Sr'
(dl Carry out the s;lme construction for spin t.
4.2-4 Determine the isospin assignments IIIJ) for each ofthe following pilrtides (refer to the
n
i Chapter 1): rr, E+, ::;0, p+, �. K!'.
4.25 (a) Che<.k that the Gell·Mann-Nishijima formub works for the quarks ", d, and •.
Eightfold Way diagrams
(b) What are the appropriate isaspin assignments, Iii]), for the allliquarks, ii, d, and i?
Ched;: that your assignment is consistent with the Gell·Mann-Nishijima formub.
Gell·Mann-Nishijima formula holds for all hadrons made out of ". d, s. ii. d. and i)
(Since Q, h, A. and S aU add. when we combine qUllrks. it follows that the
which is to say long before the disconry of charm. beauty. or truth. Vsing the table
4.2ti (al The <;ell·Mann- Nishijima fonnub, Equation 4.37, was proposed in the earlyfifties.
of quark properties (in Section 1.11), and the quark isospin assignments, Equation
4.38. deduce the gtMral formula expressing Q in terms of A, I], S, C, B, and T.
(bl Beause .. and d are the only quarks with nonzero isospin. it should be possible
to express I) in terms of U rupness') and D rdownness'). What's the formub)
Likewise, express A in terms ofthe Havor numbers V, D, S. C, B, and r.
(el Putting it all together, obtain the formula for Q in terms ofthe Havor numbers (that
is, eliminate A and I] from your formula in pilrt (a)). Ibis final version represents
the cleanest statement ofthe Gell-Mann-Nishijima formula. in the three-generation
quark model.
i
4.27 For two isospin- pilrtides. show that IILI.,III _ ! in the triplet state and -� in the
singlet. (Hilll: I... == ,(II ,(2); square both sides.)
+
us (al Referring to Equations 4.47 and 4.48. work out all the fI N scattering amplitudes. -Ii.
through -lij. in terms of -Ii, and -Iil .
(bl Generalz
i e Equation 4.49 to include all 10 cross sections.
(el 'n the same way, generaJize Equation 4.50
t channel dominates: (i) "'- + P -+ J<'l + EO; (b) fI- + p ..... K+
+ E-; (c) fI+ + P -+ K+ + E+. Whit if the energy si such that the I = i channel
4.29 Find the ratio ofthe cross sections for the following reactions, assumn
i g the eM energy
is such that the I =
•.30 What are the possible total isospins forthe following reactions: (i) K- +p -+ EO + flO;
dominates?
(b) K- + p
...... E+ + It-; (c) K!' + p ...... E+ + ito; (d) "fl!! + p -+ 1:0 + ".+. Find the ratio
ofthe cross sections. assuming one or the other isospin channel dominates.
4.31 On the graph in Figure4.6, we � 'resoninces' as 1525, 1688, 1920, and 2190 (is well as
the on� at 1232). By comparing the twocutves, determine th� isospin oreach resonanc�.
i. and I!>. for any
t. Thus the nucleon is N(939). ind the 'original' I!>. is 1!>.(1232)_ Name the
The nomenclature is N (followed by the m�ss) for any state with 1 ==
state with I =
other resonances, and confirm your answers by looking in the Parikh Physics Bockla.
4.32 The 1:.0 can deciY into E+ + "'- , 1:0 + flO, or E- + fI+ (aJso A + flO, but we're not
concerned with that here). Suppose you observed 100 sllch disintegrations. how many
would you expe.::t to � ofeich type?
33 (al The or particle is a bound state of two protons and two neutrons. that is. a �He
•.
nucleus_ There is no isotope ofhydrogen with in atomic weight of four ('H), nor of
lithium 'Li. What do you conclude about the isaspin ofan or pilrtide?
(blThe reaction d + d ...... or + flO has never been obse!"\·ed. Explain why.
(el Would you expect 'Be 10 exist? How about a bound state offour neutrons?
4.4 Oi5ueU Symmelrie3
4.34 (a) Using Equation 4.52. prove thatthe eigenvilues of P ar� ±1.
(b) Show Wtany scalarfunctionfIx. y, z) canbe expressed as the sum ohn eigenfunction
f+(x. y. z) with eigenvalue +1 and an �igenfunctionf (x y, z) with eigenvalue -1.
Construct the functionsf+ andf in terms off. IHilll: Pfix. y. z) f(-x. - y, _
,
_,
=
z).)
us (a) Is the neutrino an eigensl<lte of P? Ifso. what is its intrinsic parity?
("') Now Wt � know f
+
and ()+ are actually both the K+, which of the decays in
Equation 4.53 actually violates p,arity conS�IVation?
i formation in Table 4.6, determine the C p,arity ofthe following mesons:
4.16 (a) Using the n
".,p.w,'I, '1',¢,fI ·
(b) Show that RIIIO) _(_1)1110), and use this result to justify Equation 4.60
4.37 The dominant decays ofthe 'I meson are
'1 _ 2y(39%),
'1 - 3:11" (55%),
'1 - :1I":1I"y(5%)
(4.78)
and it is classified as a 'sl<lble' p,article. so evidently none of these is a purely strong
at 54<) MeV/c!. the '1 has plenty of mass to
deoy strongly into 2". or 3"..
[a) Explain why the 211" mode is forbidden, for both strong and electromagnetic n
i terac·
interaction. Oflband, this s�ms odd, since
tions.
1"'1 Explain why the 311" mode is forbidden as a strong interaction, but allowed as an
elearomOlsmfic decay.
4.38 For two hadrons to int�rconvert, A"",B, it is necessary that they have the same mass
(which in practice means that they must be antiparticles of one another), the sam�
chuge, and the same 1».ryon nwnber. In the Standard Model. with the usual three
generations, show that A and B would have to be neutral mesons. and identifY their
possible quark contents. What, then, are the candidate mesons? Why dOC'sn't the neutron
mix with the antineutron. in the same way as the � and f<'l miJc to produce Kl and Kl?
Why don't we see mixing ofthe n�utral strange l'tdor mesons Jo:<" and J(O-?
09 Suppose you wanted to inform someone in a distant galaxy that humans have their
hearts on the left side. How could you communicate this unambiguously, without
sending an actual ·handed' object (such as a corkscrew, a circularly polarized light beam.
or a neutrino). For all you know their galaxy may be made of antimaner. You cannot
afford to wait for ..ny replies, but you are allowed to use English.
4.010 The charged weak n
i teractions couple a d. 5, or b to a u. ,. or t. but a d (for example)
unnot go directly to an 5 or a b. However. such a coupling can occur indirectly, vii a
so-called "penguin·' diagram, in which a quark �mits a virtual W that it subsequently
reabsorbs. having in the mean time interacted with a gluon:'
A
'
b
,
W
d
A ·tree· diagram is onewith noclosed loops. Construct a penguin diagram representing
80 -+ 11" '" + ".-. and a tr� diagram for the same process (the ia"er should have no
gluons). In ooth cases, let th� d quark � a spectator. ['Direct' CP violation comes from
the interference of these two diagrams.]
Don·tlook for anything resembling the bird here - the IUIm is a joke. The SI(l'1' is told bnt by
Woit. P. (20(6) NOl E_ Wrong. Buk Books, New York, pp. 5-4-55.
1 157
5
Bound States
Thefirstpart ofthis chapter is devoted to the nonrtlativistic theory oftwo-partick bound
states - hydrogen (t-p+), positronium (e-c+), tharmonium (cC), alia bott{)monium
(b'b). This material is not ustd in subsequent chapters and call be skimrm:d, sallcdfor
later, or slo:ipptd entirely. Some �tlaintanu with quantum muhanics is essential. TItt:
final two sections (5.5 and 5.6) conam relativistic light.quark systems - the familiar
mesons and baryons - about whichfar Itss can be said with conjidena. I conctntrate on
the spin/flavor/color structure oftJu wavefonctums and devtlop a modelfor tstimating
masses and magntti<.: moments,
5.1
The SchrMinger Equation
The analysis ofa bound state is Simplestwhen the constituents travel at speeds sub·
stantially less than c, for then the apparatus of nonrelativistic quantum mechanics
can be brought to bear. Such s
i the case for hydrogen and for hadrons made out of
heavy quarks (e and b). The more familiar light-quark states (made out of u, d, and
s) are much more difficult to handle, because they are intrinSically relativistic, and
quantum field theory (as currently practiced) is not well suited to the description of
bound states. Most ofthe techniques available assume that the particles are initially
free, and become free again after some briefinteraction, whereas ina boundstate the
particles interact continuouslyover an extended period. Thus there exists a very rich
theory of ,charmonium' (ee, the 'It meson system), and 'bottomonium' (bb, the "]
system), but comparatively little can be said about the excited states of uii (say) or ad.
i relativistic or not? The sim·
How can you tell whether a given bound state s
i small compared to the rest
plest criterion is as follows: if the binding energy s
energies of the constituents, then the system is nonrelativistic." For example,
• In gene",l. the total energy of a composite
sys�m is the sum ofthr« terms: (i) the
rest �nergy of the constituents, (ii) the ki·
netic energy oflhe constituents, and (iii) the
potential energy ofthe configuration. The
J..tter two are typically compo.rable in si""
(the precise relation is given by the virial
than the constituent res! energies. so too is
their kinetic energy. and hence the system is
nonrelativistic. On the other hand. if the mass
of the composite structure is substantially
different from the sum of the rest mas!Oe$ of
the constituents, then the kinetic energy is
large and the system is reJ..tivislic.
theorem). If the binding energy is much less
IntrodllClion I<l Eltm.:nl<ll)' P�01;dts, St",,,,d Editi.".. David Griffith.
Copyright (I 2008 WILEY·VCH Verlag GmbH & Co. KGaA. Wei.nheim
ISBN: 978-J.S27--40601·2
160
I 5 80und Slotts
the binding energy of hydrogen is 13.6eV, whereas the rest energy of an elec·
tron s
i 5 1 1 000 eV - this is dearly a nonrelativistic system. On the other hand,
quark-quark binding energies are on the order of a few hundre<! MeV, which is
about the same as the effective rest energy of u, d, or s quarks, but substantially
less than c, b, and t (see Table 4.4). So the light-quark hadrons are relativistic, but
heavy-quark systems are not.
The foundation for nonrelativistic quantum theory is Schrodinger's equation [1 J.
( 2m
)
h' , + V
--v
iH
<11 = 1.h' <II
(5 .1)
[t governs the time evolution ofthe wavefonction \jI(r, I), describing a particle of
mass m in the presence ofa specified potential energy Vir, I). Specifically, ll./I(r, t)12
dlr is the probability of finding the particle in the infinitesimal volume dJr, at time
t. Since the particle must be somewlurt, the integral ofll./l12 over all space hasto be 1:
j
l>l'12 dJr = 1
(5.2)
We say that the wave function is 'nonnalized'.'
If V does not depend explicitly on t, the SchrOdinger equation can be solved by
separalion ofvariables;
where 1/r satisfies the lime-indtptndtnt Schrodinger equation
(5.3)
(5.4)
and the separation constant E is the energy ofthe particle. The operator on the left
is the Hamiltonian;
(5.5)
and the (time-independent) SchrOdinger equation has the form of an eigenvalue
equation:
'it is an eigenfunction of H, and E s
i the eigenvalue.t
(5.6)
• A S(llution !O th� Schrodin�r equation Gin j,. multipli.-d by any constant and remain a solu·
tion. In pnctiCl', � fuc this cow;�nt by d�manding that Equation 5.2 j,. :<atisfi.-d; this proc�
is GllIrd ·nonnalizing· thr wa..e function.
t Notic� tru.t 1"'11 = It II . For most purpose; it is only th� absolut� squa� of th� wa"" function
that matters. and � smoll work almost �"'lusi�ly with t. Casually. we often refer to ¥r as 'the
w.� function·, but �m�m� that the aawa! wave function carries the aponential ti� �n·
dence.
5. 1 Th� Xhrllding�r Eql'otion
Table 5.' Spherical h�rmonics for I = O. 1. 2. ilnd 3
yg = 1
r= '
,"
Y:
'11 =
y{3
"; cosO,
y[T
s,; sin{)r!I. Y:
= _
= _
(fS sin{)c05{)�
ys,;
' [w;0S . l
Yl
=
'
".
- Sin {)cosO�
[n the case of a spherically symmetrical (or 'central') potential, V is a funcion
t
only of the distance from the origin, and the (time-independent) Schrodinger
equation separates in spherical coordinates:
(5.7)
Here Y is a sphuical hanncnic; these functions are tabulated in many places
(including the Panick Physics BooHrl): a few of the more useful ones are given in
Table 5.1. The constants I and IfIj correspond to the orbital angular momentum
quantum numbers introduced in Chapter 4. Meanwhile u(r) satisfies the radial
Schrlidingu equation,
]
",2 d2u
",2 1(1 + 1)
+ V(r) + - -- u = Eu
2m dr2
2m r2
---
[
(5.8)
Curiously, this has exactly the same form as Equation 5.4 for one dimension, except
that the potential s
i augmented by the untrifogal ba""ur, (",2/2m)I(1 + 1)/r2 .
That is about as far as we can pursue the matter in general terms; at this point we
have to put in the particular potential VIr) forthe problem at hand. The strategy is to
solve the radial equation for u(r), combine the result with the appropriate spherical
harmonic, and multiply by the exponential factor exp(-iEt/Ii), to get the full wave
function "'. In the course of solving the radial equation, however, we discover that
only certain special values of E lead to acceptable results. For most values of E the
solution to Equation 5.8 blows up at large r, and yields a non-normalizable wave
function. Such a solution does not represent a possible physical state. This rather
1 '6'
1621 5 Bound
Slates
technical detail is the source ofthe most striking and important feature ofquantum
mechanics: a bound state cannot have just any old energy (as it could classically);
instead, the energy can take on only certain specific values, the (lUowed
tne� of
the system. Indeed, our real concern is not with the wave function itself, but with
the spectrum of allowed energies,
5.2
Hydrogen
The hydrogen atom (electron plus proton) is not an elementary particle, of course,
i that of the
(relatively) that it just sits at the origin; the wave function in question s
but it serves as the model for nonrelativistic bound systems. The proton is so heavy
eketron, Its potential energy, due to the electrical attraction of the nucleus, is (in
Gaussian units)
�
(5.9)
,
VIr) = -­
When this potential s
i put into the radial equation, it is found that normalizable
solutions occur only when E assumes one ofthe special values
Z .l(2nl ) ""
' 2 = -a rrn;E� = --
me4
21i. n
2
,
-13.6 eV/n
(5.10)
where n = I, 2, 3, . . . , and
Q' E
�
1
he "" 137.036
(5.11)
is the fine structure constant. The corresponding (normalized) wave function,
�...I..... (r, (i , I/! , � , is
(-,-)
[ '
na
where
a ",
' )'"
(n - 1 - 1).
2n((n + I)!jl
,_rl""
.'
"" 0.529 x 10-8 em
mel
( ,, ) L21+1 ( ,, )
na
,
�-I-l
na
ym'I' ¢),_iE.'/A
!
'
(5,12)
(5.13)
Bohr rudius (roughly speaking, the size of the atom), and L is an associated
Laguerre polynomial.
is the
Obviously, the wave function itself is a mess. but that's not really what concerns
us. The crucial thing is the formula of the allowed energies, Equation 5.10. It
was first obtained by Bohr in 1913 (more than a decade before the Schrodinger
equation was introduced) by a serendipitous amalgam of inapplicable classical
5,} H)'drogen
ideas and primitive quantum theory - an inspired blend, as Rabi put it, of'artistry
and effrontery'.
Notice that the wave function is labeled by three numbers: n (the principal
quantum number), which can be any positive integer - it determines the energy of
the state (Equation 5.10); I, an integer ranging from 0 to n - 1 that specifies the
total orbital angular momentum (Equation 4.2); and �, an integer that can assume
any value between -I and +1, giving the z component of the angular momentum
(Equation 4.4). Evidently, there are 21 + 1 different m/'s, for each I, and n different
I's, for each n. The total number of distinct states that share the same principal
quantum number n (and hence the same energy) is, therefore
(5. 14)
This is called the dtgtneracy of the nth energy level. Hydrogen is a surprisingly
degenerate system: spherical symmetry alone dictates that the 21 + 1 states with a
given value ofthe total angular momentum should be degenerate, since they differ
only in the orientation of l, but this suggests a sequence 1, 3, S, 7, . . . , whereas
the energy levels of hydrogen have much higher degeneracies: 1, 4, 9, 16, . . . . This
s
i because states with different I share the same n; it is an unusual feature of the
Coulomb potential.
In practice, we do not measure the energies themselves, but rather the wave­
length ofthe light emitted when the electron makes a transition from a higher level
to a lower one (or the light absorbed when it goes the other way) 12). The photon
carries the difftrtna in energy between the initial and final states. According to the
Planck formula (Equation 1.1),
(5.15)
The emitted wavelength is therefore given by
(5.16)
where
�.
, = 1.09737 x lOsIcm
.. 4JT ti C
(5.17)
Equation 5.16 is the famous Rydberg formula for the spectrum of hydrogen. It
was discovered experimentally by nineteenth-century spectroscopists, for whom
R was simply an empirical constant. The greatest triumph of Bohr's theory was
its derivation of the Rydberg formula, and the expression for R in terms of the
fundamental constants m, t, c, and Ii (Figure 5.1).
1 '63
164
1 5 Bou"d StIIles
�"nh.d.... __
..
"
"
,
"
..
l'
••
•
•
•
I
f
!
t
i
•
•
t
i1
I )
•
•
•
,
,
L ...- __
•
Fil' 5.1 Th� spKtrum of hydrogen. When
�n �tom changes from one sute to �n·
othtr, th� difft�nct in tnergy app"'� 1$
a quantum of radiation, Tht energy of the
photon is directly proportional to the fre·
qutnl)' oftht radiation and in��rsely pro­
portional to its wa�elength. Absorption of
radiation stimulates a trans ition to a stat�
of higher energy; an atom falling to a stile
,
of Io_r en�rgy emih radiation. Th� spec­
trum is o.g�nized into series of li nes thaI
share a common l o_r 1�1. Wiv�l�ngths
are gi�en in angstroms: the relative inten.
sity of th� lines is indicated by thkknen.
{Source: Hansch, T. W., Schawlow, A. L. and
S�riu, G. W. (Much 1979) 'The Spectrum of
Atomic Hydrogen', Sti'"lili' Ameri""", p_ 9<4,
reprinted by p".mission.)
5.2 Hydroge"
5.2.1
Fine Structure
As the precision ofexperimental spectroscopy m
i proved, small departures from the
Rydberg formula were detected. Spectral lines were resolved into doublets, triplets,
and even larger families of closely spaced peaks. This fim structure is actually
attributable to two distinct mechanisms:
i the Hamiltonian
1. Rtlativi5tic correction: The first term n
(Equation 5 .5) comes from the classical expression for kinetic
energy (p2/2m), with the quantum replacement p --j. -ih'il.
The lowest-order relativistic correction (Problem 5.4) is
_p4/8m1c2 .
2. Spin-orbit coupling: The spinning electron constitutes a tiny
�' = -ItICS,
magnet, with a dipole moment'
(5.18)
From the electron's perspective the 'orbiting' proton sets up
a magnetic field B, and the spin-orbit term is the associated
magnetic energy -�.·B.
(
-1)
The net result is a perturbation ofthe nth energy level by the amount [1J
6Ef:• = -(J tncl-4
,
1
4n
l
'.
-- -
U + !)
2
(5.19)
where j = I ± is the total angular momentum (spin plus orbital) ofthe electron
(Equation 4.12). Recall that the Bohr energies go like (J2 mc2 (Equation 5.10); fine
structure carries two more powers of (J, and hence is smaller by a factor of about
10-·. So we're talking about a tiny correction.t Since I can take on any integer value
in the late 1940s. By now. both experimental
, In W SI syst..m th� =gn�ti( dipol� momffit
and theoretical determinations ofth� aIWma­
is defined u current times �re� ('/I), but in
lollS m<IgIItti<: mome,,1 of the electron have
Caussian units it is //llc. The proportionality
been carried to fantastic precision, and stun­
factor �n the =gnnic dipole mom�nt
ning agr�ment {4].
�nd the anguLu momentum is known u the
t Thejin. SltIlClUrt "'''1111'''. (I!. o�s its name
gyromo.gneli< rWio. Classically. it should h;we
to the f�t that it {or rather. (l!l) sets the rel­
the (Gaussian) valu� (3) -'l2wot;, and this is
ative scale of th� fin� strllCtur� in hydrogen.
correct for orbillll angular momentum. But it
Howev�r. one might equally well say that (11
turns out that spin is "twice as effective as it
sets the scale of the Bohr levels themselves.
ought to be' in producing a magnetic dipole
Actually, th� best w.y to characterit� th�
(on� of the maior successes of Dirac's original
fine structure constant is to uy that it is the
theory of the electron was its explanation of
this extra 2). As it happens. however. even !his dimensionless meuure (in units of Ii<) of
the {square of the) fundamental charge: (I! '"
is not quile right: ther� are minUI� COlIt<:­
tions introduc�d by quantum dNtrodyn;tmiCS
,}fllr_
(QED) that were first calculated by Schwinger
1 165
0 · ' __
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
___
,. .
'"
,. ,
'"
,. ,
'"
,. ,
'0'
Fig. 5.2 Fine structure in hydrogen. The
rrth Bohr level (nne line) splits into n sub·
levels (dashed lines). characterized by j =
1. ! . . . . . (n - !l. hcept for the last of
these, two different values of I contribute to
"- - - - I � �
I and ! '"j + t. Spectra­
scopis�' nomMdalure - S for I = 0, P for I
= 1, 0 for I = 2, F for I = 3
is indiClited.
All I�els are shifted downward, as shown
(the diilsram is not to sule, however).
��cn lev�l: I ""j -
from 0 to n - 1.) can be any half·integer from
EK• splits into n sublevels (see Figure S.2).
_
l to n - i; thus the nth Bohr level.
5.2.2
The Lamb Shift
A striking feature of the fine structure formula (Equation 5.19) is that it depends
only onj. not on I; in general. two different values of I share the same energy. For
example, the 2SI!l (n = 2, I = O,j =
and 2PI/1 (n = 2, I = l.j =
states remain
perfectly degenerate. In 1947, Lamb and Retherford performed a classic experiment
[5] which demonstrated that this is not, in fact, the case; the S state is slightly
higher in energy than the P state. The explanation of the Lamb shift was provided
by Bethe, Feynman, Schwinger, Tomonaga, and others; it is due to the quantization
ofthe declromagneticjidd itslif. Everywhere else in the analysis - the Bohr levels,
fine structure fonnula, and even hyperfine splitting (in the next section) - the
electromagnetic field is treated entirely classically. The Lamb shift, by contrast, is
an example of a radiativt correction in QED, to which the semiclassical' theory is
insensitive. In the Feynman formalism, it results from loop diagrams, such as those
in Figure 5.3, which we shall discuss quantitatively later on.
!l
lJ
• I call it St"'�ssical l>eause !be eltaro� is �lled <1�nlUm medumially. wherels
rruogneticfitld is trealed elissiaUy.
the eltt!r().
5.2 Hydrogen
•
•
•
,
•
,
,
,
Vacuum polariUlion
,
Electron m." renorm.li181ion
,
Aoomalous migneti.:: moment
Fig. S.l Some loop diagrams contributing to the Lamb shift.
Qualitatively, the first diagram in Figure 5.3 describes spontaneous production
of electron-positron pairs in the neighborhood ofthe nucleus (misnamed \It1cuum
polarization), leading to a partial screening of the proton's charge (Figure 2.1).
The se.::ond diagram reflects the fact that the ground state of the ele.::tromagnetic
field is not zero (6J; as the electron moves through the 'vacuum fluctuations' in
the field, it jiggles slightly, and this alters its energy. The third diagram leads
to a tiny modification of the electron's magnetic dipole moment (see footnote to
Equation 5.18). We are not in a position to calculate these effects now, but here are
the results {7J: For I 0,
""
"
,
Ll.Er...mb "" ex me ) k(n,O)
"
(5.20)
.
where kIn, 0) is a numerical factor that varies slightly with n, from 12.7 (for n = 1)
to 13.2 (for n ...... 00). For 1 =1= 0,
"
'
Ll.Et...mb "" a me 4n)
I
k(n, � ±
.
l
'
I
1!{j + 1)(1 + I)
)
'
forj = I ±
!
(5.21)
where kIn. � is a very small number (less than 0.05) that varies slightly with n and I.
Evidently the Lamb shift is miniscule, except for states with I "" 0, where it is about
one·tenth the size of the fine structure. However, because it depends on I, it lifts
the degeneracy of the pairs of states with common n and j, in Figure 5.2, and in
particular it splits the 251/2 and 2Pl/2 levels (Problem 5.6).
5.2.3
Hyperfine Splitting
Fine structure and the Lamb shift are minute corrections to the Bohr energy levels,
but they are not the end of the story; there is a refinement that s
i smaller still
i of the nucleus. The proton, like the electron,
(by a factor of 1000), due to the spn
1 167
168 1 5 Bound 5totes
structure and lamb sMt)
En (modified by fiflEl
Triplel
/:I
Fig. 5.4 Hyperfine splitting for I
..
Singlel
0.
constitutes a tiny magnet, but because it s
i so much heavier, its dipole moment is
much smaller:
(5.22)
(The proton is a composite object, and its magnetic moment is not simply th./2mpc,
as it would be for a truly elementary particle of spin !. Hence the factor Yp'
whose experimental value is 2.7928. Later on we shall see how to calculate this
quantity in the quark model.) The nuclear spin interacts with the electron's orbital
motion by the same mechanism as the spin-orbit contribution to fine structure; in
addition, itinteracts directlywith the electron spin. Together, thenuclear spin-orbit
interaction and the proton-electron spin-spin coupling are responsible for the
hyptrjine splitting [8]:
6£ f
h
¥,
±!
-- (-m ) a�mc2"-�;=;
'--T for ! = j ± t
2nJ if + })(I + l)'
mp
(5.23)
where! is the kJlll! angular momentum quantum number (orbital plus bothspins).
Comparing the fine structure formula (Equation 5.19), we see thatthe difference
in scale is due to the mass ratio (mlmp) out front; it follows that hyperfine effects
in hydrogen are about 1000 times smaller. If the orbital angular momentum is
zero (I = 0), then! can take on two possible values: zero, in the singlet state (when
the spins are oppositely aligned) and one, in the triplet state (when the spins are
parallel). Thus, each I = 0 level splits into two, with the singlet pushed down and
the triplet lifted up (Figure 5.4). In the ground state [9] n = 1 the energy gap s
i
(5.24)
corresponding to a photon ofwavelength
2.",
, = -- = 21.1cm
,
(5.25)
This is the transition that gives riseto the famous '21-centimeter line' in microwave
astronomy PO].
S.J Posilronium
5.3
Positronium
The theory of hydrogen carries over, with some modifications, to the so-called
'exotic' atoms, in which either the proton or the electron is replaced by some other
particle. For instance, one can make muonic hydrogen (P+M-), pionic hydrogen
(p+]f -), positronium (e+e-), muonium (M+e-), and so on. Of course, these exotic
states are unstable, but many of them last long enough to exhibit a well·defined
spectrwn. In particular, positronium provides a rich testing ground for QED. It was
analyzed theoretically by Pirenne in 1944, and first produced in the laboratory by
Deutsch in 1951 [llJ. In particle physics, positronium assumes special m
i portance
as the model for quarkonium.
The most conspicuous difference between positronium and hydrogen is that we
are no longer dealing with a heavy, essentially stationary nucleus, around which
the electron orbits, but rather with two particles of equtll mass, both orbiting the
common center. As in classical mechanics, this two-body problem can be converted
into an equivalent one-body problem with the reduced mass [1 J
In the case of positronium ml = m2 = m, so Innd = m/2, and we get the
unperturbed energy levels by the simple substitution m ..... m/2 in the Bohr
formula (Equation 5.10):'
(S.2?)
For example, the ground-state binding energy s
i 13.6eV/2 = 6.8eV. The wave
functions are the same as hydrogen's (Equation 5.12), except that the Bohr radius,
which goes like 11m (Equation 5.13), is doubled:
aPO' = 211 = 1.06 x 10-
8 cm
(5.28)
The perturbations run much as before, apart from pesky numerical factors, with
one dramatic exception: in positronium, the hyperfine splitting is ofthe same order
as the fine structure (a4mc2 ), since the mass ratio (m/mp) that suppresses proton
spin effects in hydrogen is one for positronium.t Meanwhile, since the 'nucleus'
(t+) is no longer stationary, there is a new correction due to the finite propagation
•
In the c.ose ofhydrogen, the reduced tnlSS differs from the dtclron tnl5S by only a very srnill
amount. about O.OS%. Nevertheless. ttchnically the lit in the Bonr formula is the reduced nuss,
and this leads 10 oo�",.ble differencn lJ,:,lwttn the spectra of hydrogen and deuterium_
t This leads 10 some termin<>logic.ol confusion in the literature. I'D use the words 'fine structure'
for aU perturbations of order a'""l. except the pair annihilation term (�below). including the
spin-spin and positron spin-orbit couplings, wh� aruo.logs in hydrogen would be called 'hy-
p...fi",,·.
1 169
170
I 5 Bound Slatf:5
,
,
,
,
Fig. 5.5 Pair �nnihilation diagram, wflich affe<ts the spec·
trum of positronium but d�s not OCCur in hydrogen.
time for the electromagnetic field; its contribution is also oforder (a 4mcl). When all
this is put together, the fine structure formula for positronium is found to be [11]
Ef"" - a 4 mc2 _
_
'
2nl
f.
[�_ H ]
32n
(1 +
(21 + 1)
(5.29)
where E = 0 for the singlet spin combination, whereas for the triplet'
-(3/ + 4)
(1+ 1)(21 + 3) '
,=
forj = l +
forj = 1
,
1(/ + 1) '
(31- 1)
/(21 - I) '
l
(5.30)
forj = I - I
The lamb shift, of order aSmc2 , makes a smaUish correction to this; however,
since the 'accidental' degeneracy is already broken at the fine structure level n
i
positronium, this contribution loses much of its n
i terest. There is, however, an
entirely new perturbation, with no analog in hydrogen, resulting from the fact
that e+ and e- can annihilate temporarily to produce a virtual photon. In the
Feynman picture, this process is represented by the diagram in Figure 5.5. Because
it requires that the electron and positron coincide, this perturbation is proportional
to IIJI (0)12, and hence occurs only when I = 0 (<l> goes like ,.l near the origin - see
Equation 5.12). Moreover, since the photon carries spin 1, it takes place only in
the triplet configuration. This process raises the energy ofthe triplet 5 states by an
amount
.
"
D.E�n = a me
l
4nJ
(5. 31)
(1 = 0, 5 = 1)
• In hydrogen, where the prolOn spin IS,) COn_
tributes on y �t the hyperfine level. we used
J for the sum of the electron's spin and or·
bital angular momentum a = l + S,); for the
IOta] angular momentum we needN a new
letter. F = l + S, + S,. In pos;tlonium the
spins contribute on an equal footing. and
it is custoJt»ry to combine them first (S
S, + S,) and use , for the total: J = L + S, +
two
"
=
SA Q�Q,*oni�m
- the same order as fine structure. The complete splitting of the n = 1 and n = 2
Bohr levels in positronium is indicated on Figure (5.6)'
As in the case of hydrogen, positronium can make transitions from one state
to another with the emission or absorption of a photon, whose wavelength is
determined by the difference in energy between the two levels. UnUk.t hydrogen,
positronium can also disintegrate completely, the positron annihilating the elec­
tron to produce two or more real photons. The charge conjugation number for
I
positronium is (_I) +>, while for n photons C = (_I)K (see Section 4.4.2). Thus,
charge conjugation invariance prescribes the selection rule
(5.32)
for the decay ofpositronium from the state I, s to n photons. Since the positron and
electron overlap only when 1 = 0, such decays occur only from S states.t Evidently,
the singlet (s = 0) must go to an even number of photons (typically two), whereas
the triplet (s = 1) must go to an odd number (typically three). In Chapter 7 we will
be in a position to calculate the lifetime of the ground state:
,.
,= ' 2 = 1.25 )( to-tO seconds
" =
(5.33)
,.4
Quariconium
In the quark model all mesons are two·particle bound states, qtq2. and it is natural
to ask f
i the methods developed for hydrogen and positronium can be applied
to mesons as well. Ught-quark (u, d, 5) states are intrinsically relativistic. so any
analysis based on the Schrodinger equation is out ofthe question, but heavy-quark
mesons (cc, be, and bb) should be suitable candidates. Even here, however, the
interaction energy (E) is such a substantial fraction ofthe total that we are disposed
to regard the various energy levels as representing different parlidts, with masses
given by
(5 .34)
Unlike hydrogen and positronium, n
i which the forces at work are entirely
electromagnetic, and the energy levels can be calculated to great precision, quarks
are bound by the strong force; we don't know what potential to use, in place of
Coulomb's law, or what the strong analog to magnetism might be, to obtain the
spin couplings. In principle, these are derivable from chromodynamics, but no one
Positronium states a� convnllionally labd..-d n("+'I�. with I given in spectr()$(opisfs notltion (S
for I '" O. P for I "" 1. D for I "" 2. etc.). md I the IOtll spin (0 for the singlet, 1 for the triplet).
t ACluilly, positronium can in principle dec."iy directly from a Stile with I > 0 by <I higher-<lrner
process. but it is much more likely to cascade down to:an S st:ate first, :and decay from the�.
•
1 1n
'''''''
000
"'"
00
'
3 >S,
[
Cflarmoolum
t
7
•
•
.l
,
!
!
�
. .. �-----=====-------------- �
� 12
,t
500
•
,
\
.
Sol�cl
2'S,I,,')
-:.'�-
...
"'"
l . ,.. , 11(, 1
2 ' Po IXo)
,.,
' 00
1 '5,(.,,)
,
- '00
Z 'P1IX,1
'50111..
�
f
�
<
'1',..,..
Posltronlum
Oiaocil,kln
energv
2 's,
�H
;::
•
2 '1',
"
" '. \/;:
,:=
: =
2'1'
2 'SO
2 '1"• /
,
I
".,
\ 1 !O:
�
x
1 's,
1 'So
__
'P,t
....
'S'II'"
Fig. 5.6 S�ctrum of �nergy levels in positronium and ch..rmonium. Note that the scal., is
gruter by a f"ctor of 100 million for charmonium. In positronium, the various combina­
tions of angular momentum cause only minuscule shifts in energy (shown by expanding the
vertical scale), but in charmonium the shifts ;u., much larger. All energies are given with
reference to the l 1S1 state. At 6.lIeV positronium dissociates. At 633 MeV a� the energy
of the '" (harmonium becomes quasi·bound, because it Can dear into r::fl and FJl mesons.
{Source: Bloom, E. �nd Feldm�n, G. (M�y 1982) 'Quarlconium'. Scienlific American, p. 66,
rep<int� by permission.)
"
,
o
1 's,,(�cl
'5 0111..
•
�
�
'SOl....
'I' 'lit..
'1' 'lit..
�
�
�
,
5.4 Quorlronj�m
Table 5.2 'Bohr' Energy levels for linear-plus-Coulomb poten­
tial (Equation 5.83) with various V.IUM of Fo. They are for
S-states (I = 0) and assume a, = 0.2. m = 1500 MeV/c2 (re­
duced mass, 750 MeV/C:-).
5\ (MeV)
Fo (MeVfm-t)
5� (MeV)
5J (MeV)
5. (MeV)
500
)07
677
%1
1210
tOOO
533
727
1100
1480
1550
2040
1940
2550
1500
Nunu.rkal results from unpublished tables p"'pared by Richard E.
Crandall.
yet knows how to do the calculation. Still, we can make some educated guesses,
for chromodynamics is very similar in structure to electrodynamics, except for
some nonlinear terms which, in the light of asymptotic freedom, probably don't
contribute much at short distances_
In quantum chromodynamics (QeD), the short-distance behavior is domi·
nated by one-gluon exchange, just as in QED it is dominated by one-photon
exchange. Since the gluon and the photon are both massless spin-I particles,
the interactions are, in this approximation, identical, apart from the overall cou­
pling strength and various so-called 'color factors', which result from counting
the number of different gluons that contribute to a given process. At short
I/r, and a fine
range, therefore, we exPe(t a Coulomb-like potential, V
structure that is qualitatively similar to that of positronium (121. On the other
hand, at large distances we have to account for quark confinement: the po­
tential must increase without limit. The precise functional form of V(r) at
large r is rather speculative: some authors favor a harmonic oscillator po­
tential, V ....., r2, others a logarithmic dependence, V ...... In (r), still others a
linear potential. V � r, corresponding to a constant force, The fact is, any of
these can match the data reasonably well, because they do not differ substan­
tially over the rather narrow range of distances for which we have sensitive
probes.
For our purposes, we may as well choose
�
V(r) =
-
J ,
4 a,fic
-
-
+ For
(5.35)
where a. is the chromodynamic analog to the fine structure constant, and j is the
appropriate color factor, which we'll cakulate in Chapter 8. Unfortunately, exact
solutions to the Schrodinger equation with linear-pius-Coulomb potential are not
known, and I cannot give you a simple formula for the 'Bohr' energies. However, it
can, ofcourse, be done numerically (see Table 5.2), and Fo can then be chosen so as
to fit the data (131 (Problem 5.11). The result is about 16 tons(!), or, in more sensible
units, 900 MeV fm- I, which is to say that a quark and an antiquark attract one
1 173
5 Boul1d Slalef
another with a force of at least 16 tons, regardless of how far apart they are.' This
perhaps makes it easier to understand why no one has ever managed to pull a free
quark out of a meson.
SA.1
Channonium
Shortly before the discovery of the "" Appelquist and Politzer [141 suggested that
if a heavy 'charm' quark existed (as Glashow and others had proposed) it should
form a nonrelativistic bound state, ce, with a spectrum of energy levels similar to
positronium. Theycalled the system 'charmonium' (which does more to emphasize
the parallel than to beautify the language) . When the '" was found, in 1974, it was
quickly identified as the l' 5l state of charmonium.t (In the SLAC experiments
the 1/1 was produced from t+t- annihilation through a virtual photon: t+ e- ...... y
-+ 1/1, so it has to carry the same quantum numbers as y - in particular, spin 1.
Thus it could not be the ground state of charmonium, but presumably it was the
lowest-lying state with total angular momentum 1.) Consulting the positronium
energy-level diagram (Figure 5.6), we immediatelyanticipate a spin·O state at lower
mass (the 11 So) and six n = 2 configurations. Within two weeks the 1/1' (235 ) was
1
found. This was easy, because it again carries the same spin - and parity - as the
photon; it was produced n
i the same way as the 1/1, simplyby cranking up the beam
energy.
In due course all the n = 1 and n = 2 states were discovered [151, save for the 21 PI
at a predicted mass of about ]500 MeV/c?-, which presents special experimental
problems. The following nomenclature has been adopted: singlet 5 states (spin 0)
,
are called 'I< s. triplet 5 states (spin 1) are 1/I's, and triplet P states (spin 0, 1. or 2)
are designated XcO, X,b Xa. For a while the value of n was indicated by primes,
but this quickly got out ofhand, and the current practice is simply to list the mass
parenthetically; thus for n = 1 we have ", = 1/1 (3097); for n = 2, 1/1' = 1/1 (3686); for n
= ], 1/1" = >It (4040); for n = 4, 1/1'" = '" (4160); and soon.* The correlation between
states ofcharmonium and those ofpositronium is almost perfect (Figure 5.6). Bear
in mind that the gap between the two n = 1 levels (which would be called hyperfine
splitting in the case of hydrogen) s
i greater by a factor of lOl l in charmonium than
n
i positronium. Yet even over so huge a change of scale, the ordering of the energy
levels and, for a given value of n, their relative spacing, are strikingly similar.
All the charmonium states with n = 1 and n = 2 are relatively long·lived,
because the OZI rule (Section 2.5) suppresses their strong decays. For n ::-: 3 the
charmonium masses lie above the threshold for (OZI.allowed) production of two
, At �xtr�mdy .hon distanc�. Fo and a, th�m�lves decr�..se. I�ading to ..symptotic freedom. but
for now we shall !reat them u constants.
t The nomenclature is borrowed from that of positronium - � foomo� after Equation 5.31.
* Some authors. including th� of the Panicle PII}'<ics Bookkl, number sta!�S conse<:utively. st..
J1·
ing with 1 for tad. combination of I. J. ..00 j. so that what J Cilll a 2P state (Figure 5.6) is listed
as lP. Sorry a"bout that. Incidentally, the 11'(3770) i. a displaced 3) V, .b.t�, and oo,.s not ruDy
�Iong in this hierarchy.
5.4 QUQ,kQn;um
�
d,
d
;;
}.,
} ._
Fig. 5.7 (i) OZI·suppressed deciY for charmonium �Iow
the DO thrtshold, (b) OZI·allowed decay for chirmonium
above the DO threshold.
charmed D mesons (0<>, DO at a mass of 1865 MeV/e2, or D± . at 1869 MeV/e2).
Their lifetimes are therefore much shorter, and we call them 'quasi·bound states'
(see Figure 5.7). Quasi·bound states of (harmonium have been observed going up
at least as high as n = 5.
5.4.2
Bottomonium
In the aftermath of the November Revolution there was widespread speculation
about the possible existence of a third.quark generation (b and t), and in 1976
Eichten and Gottfried [16] predicted that 'bottomonium' (bb) would exhibit a
hierarchy of bound states even richer than (harmonium (Figure 5.8). The bottom
analog to the D meson (to wit, the B) had an estimated mass large enough that not
only the n = 1 and n = 2, but also the n = 3 levels should be bound. In 1977 the
upsilon meson was discovered, and immediately interpreted as the 11 51 state of
bottomonium. At present, the 1 51 states have been found for n up to 6, as well as
I US
176
15
Bound Stolts
"'"
Quasi-bound ...,,,
4'S, (T"'(10579J)
"'"
Bound sto..
t
"""
"'"
""
i
�
!
i
<
, ''"
3 '5, (T"(103551l
3 'P,
-----
''''
..
..
-------
2 'So
-"'" "
--
---...
____
2 ·s, (T'II00231l
" D.
D1
3 '0,
2'P,
-----
".
""
""
, .,
,
1 'S,ITI9460lI
, ''"
-----
'5 Stitts
'P stotes
'p States
'0 st.t..
'O st.tes
Fig. 5.8 Botlomonium. Not� that th�.t a.t fa. mort tIound
stat�5 thin for chi.monium � compa.� Figu.� 5.6 (Soll'ce:
Bloom, E. and Ftldman, G. (May 1982) 'Qua.konium', Sd·
enfific AmuialM, p. 66, rtp.inttd by pt.mis$ion. (orrKlW
miS5�S from PIlr1ic/� Pkysi<:$ Book/a (2006).)
the six J P states for n = 2 and n = 3. The level spacings in the '" and T systems'
are remarkably similar (Figure 5.9), in spite of the fact that the bottom quark is
more than three times as heavy as the charm quark [17J.
,.,
Light Quark Mesons
Consider now the mesons made entirely out of light quarks (I', d, s). These are
relativistic systems, remember, so we cannot use the Schr6dinger equation, and
the theory is rather limited [lS). In particular, we shall not concern ourselves with
the spet:lrum of excited slates (Table 4.6), as we did in the case of the heavy
• In prindpl� th�r� should be a simiLor system for the B: mesonS (cb and bel, but SO far only one
of th.-se, at 6286 MeV, has bttn producN in the Lohomory.
1 JS, ------ �(3097)
o
----- T(9460)
Fig. 5.9 Level spacings in the ", and Y systMls. (Source: Particle Physics Boo/del (2006).)
quarks, but will confine our attention to the ground state, with 1 = O. The quark
spins can be antiparallel (singlet state, S = 0) or parallel (triplet state, S = 1); the
former configuration yields the pseudoscalar nonet, the latter gives the vector nonet
(Figure S.10).
To beginwith, ! want toclear upa problem that was not resolved in Chapter 1. We
obtained nine mesons simply by combining a quark and an antiquark in all possible
combinations (Section 1.8), but this left three neutral states with strangeness 0 (uii,
dd, and sSj, and it was not clear which of these was the 11"0, which the 1/, and which
the I)' (or, in the vector case, the pO, w, and ¢). We are now in a position to resolve
K'
K'
•
.'
•
.t
�,
K"
.'
,-
"
•
• •
•
w
Ve(:tOt none\
Fig. 5.10 light·quart; mesons with 1 = 0.
"
5 Bound Slates
the ambiguity. The up and down quarks constitute an isospin doublet:
(5.36)
So too do the antiquarks:
(Notice that d carries 13 =
(5.37)
+!, and u has /) = -!: within a multiplet, the particle
with the higher charge is assigned the greater fJ. The minus sign is a te<hnical
detail [19], which does not affe<t the argument here in any essential way.) When
(
we combine two particles with 1 = � , we obtain an isotriplet (Equation 4.15)
11 1) = -ud
11 0) = (uli - dCf)/..J2
11 - 1) = au
(5.38)
and an isosinglet (Equation 4.16)
100) = (uli + iJJ/..fi
(5.39)
In the case ofthe pseudoscalar mesonsthe triplet is the pion: for the vector mesons
it is the p. Evidently, the 1["0 (or the pO ) is neither uli nor dd, but rather the linear
combination
(5.40)
If you could pull a 1["0 apart, half the time you'd get a 10 in one hand and a Ii in the
other, and half the time you'd get a d and a d .
This leaves two r = 0 states (the isosinglet combination, Equation 5.39, and sS)
'
which must represent I) and I) (or w and cPl. Here the situation is not so dean, for
these particles carry identical quantum numbers, and they tend in practice to 'mix.'
In the case of the pseudoscalars the physical states appear to be
I) = (IOU + tid
- 2sS)j./6
(5.41)
I)' = (uli + dd + sS}/./3
(5.42)
whereas for the vector mesons
w = (IOU + dCf)j..fi
cP = sS
(5.43)
(5.44)
To the extent that the Eightfold Way is a good symmetry, the pseudoscalar
combinations are more 'natural', since the I)', which treats 14, d. and s symmetrically,
5.5 Lighl Quarlc Mtsons
d
Fig.
5.11
,
;;
d
Quark� and antiquarks.
i exactly the
is unaffected by SU(3) transformations; it is a 'singlet' under SU(3), n
same sense that the 11"0 is a singlet under SU(l) (isospin). The 'I, meanwhile,
transforms as part of an SU(3) 'octet', whose other members are the three pions
and the four K's. (This is, in fact, the original pseudoscalar octet.) By contrast,
neither the � nor the w is an SU(3) singlet. They are, you might say, 'maximally'
mixe<!, since the strange quark is s
i olate<! from the other two. Incidentally, the
other meson nonets seem to follow the 4> - w mixing pattern [20[.
Meanwhile, the strange mesons are constrocted by combining an 5 quark with II
"d
� = -sd.
K
-
(5.45)
In the language of group theory, the three light quarks belong to the fundamen­
tal representation (denoted 3) of SU(3j, whereas the antiquarks belong to the
conjugate representation (3) (Figure 5.11). What we have done is combine these
representations, obtaining an octet and a singlet:
(5.46)
just as n
i Chapter 4 we combined two two·dimensional (spin·l) representations of
SU(lJ to obtain a triplet and a singlet:'
(5.47)
If SU(3) were a perfect symmetry, aU the particles in a given supermultiplet
would have the same mass. But they obviously do not; the K weighs more than
three times the 11" , for example. As I indicated in Chapter 4, the breaking of flavor
symmetry is due to the fact that the quarks themselves have unequal masses; the
II and d quarks weigh about the same, but the s quark is substantially heavier.
Roughly speaking, the K's weigh more than the /l'S because they contain an 5
• Unfortun..tely (from the point of view of
notational consistency) represenutions of
SU(3) ..re custormrily bkled by their di·
�nsion. wherus representations of SU(2)
are more often identified by their spin. so
Equation 5.45 would usual!y be written as
1 0 \ '" 1 QI O. By the w..y. it ho.ppens th..t
the fund
..mental repreRnb.tion of SU(2) is
equivalent to its conjulPte: there's only one
i
kind of spn
That's why we were ahle to
t.
repr�nt ii and d in Eql1iltion 5.79 in terms
of ordinary isospin· t states. For SU(3) this is
no lon� the orR.
1 179
180
I 5 BOllmJ Slatu
Table 5.1 Pseudo scalar �nd vedor meson mass�s, {MeV/el}
,
Muon
Observed
CaJ.cubtd
K
,
,
w
K'
•
Il9
138
'"
<96
561
"g
775
7"
775
892
1031
go.
783
1020
in place ofa u or d, But, that cannot be the whole story, for if it were, the p's would
weigh the same as the H'S; after all, theyhave the same quarkcontent and are both n
i
the spatial ground state (1'1 = 1 , 1 = 0). Since the pseudoscalar and vector mesons dif·
fer only in the relative orientation of the quark spins, the difference in their masses
mustbe attributed to a spin-spin interaction, the QeD analog to hyperfine splitting
in the ground state ofhydrogen. This suggests the following meson mass formula:'
(SI . S2)
M(meson) = ml + 1'112 + A--­
(S.48)
mlm2
where A is a constant 121]. By squaring S = SI + S2. we obtain
for s = 1 (vector mesons)
for s = 0 (pseudoscalars)
)
(S.49)
For constituent masses I'II� = md = 308 MeVle2, m, = 483 MeVlel, the best-fit
value of A is (2m�/Ii)21S9 MeV/e2, and we obtain the results in Table 5.3.
5.'
Baryons
Some day. presumably, we shall be able to make nonrelativistic heavy-quark
baryons - "e, "b, ebb, and bbb. These are the baryonic relatives ofquarkonium 'quarkelium', you might call it, since the nearest atomic analog would be helium.
At present, though. it is hard enough to make a baryon with o� heavy quark, never
•
In I = 0 states the hypetfine correction is pro­
portion to the dot product of the magnetic
moments, J'I J'l: dipole momffits. in turn.
are proportional to spin anguln momffitum
and inversely proportioml to maSS. Th
..!'. the
inspiration behind Equation 5.46 Of course.
this s
i for QED, not QCD. What's worse, it
ignores the mass dependence of the Wa�
function (contained in the 'constant' A), and
il is based on nonrelativistic quantum me­
chanics. But nothing succeeds like success,
and Equation 5-'16 works surprisingly well.
(Notice, howe""r, that the �' is not included
in the table - see Problem 5_12).
S.G &'}'OII'
T�ble S." Light-quark baryons U " spin, P = p,arity, S =
strangeness, I = isospin. This is not a complete list; baryons
with spins as high as 11/2 have be-en observed)
s_
SU(3) Representa.tion
e
J
,.
,
,
10
,
r
,,
r
,
,,
!
,,
10
,
10
,
!-
,,
r
,.
,
,.
,
,.
,
r
S=O
1.0
1_ 1
S= -2
N('.J39)
A (1l16)
1: (11'.13)
S (1318)
1: (138S)
S (1530)
6 1l2J2)
S= -3
Q (1672)
1\ (1405)
A (l520)
N(153S)
1\ (1670)
1: (1610)
N(IS20)
A (1690)
A (1830)
1: (1670)
N(1675)
t.. (1620)
S (1820)
1: (177S)
t.. (1700)
N(I720)
A (1890)
N(1680J
A (1820)
1: (1'.JlS)
S (2030)
t.. (1905)
1: (2030)
6 (1950)
N(1440)
1\(1600)
1: (1660)
So","o: Ro""", ofPorlick Pkysics (2(06). S�n 14.4.
mind thru, and I won't speculate here about the heavy-quark baryon spectrum.
On the other hand, the array of observed light quark baryons is immense (see
Table S.4).
5.6.1
Baryon Wave Functions
Baryons are harder to analyze than mesons, for several reasons. In the first place.
a baryon is a three-body system. "iliere's not just OIU orbital angular momentum
to consider, but two (see Figure 5.12). We'U concentrate on the ground state, for
which I = I' = O. In that case, the angular momentum of the baryon comes entirely
from the combined spins of the three quarks. Now. the quarks carry spin
so
each can occupy either oftwo states: 'spin up' (t) or 'spin down' 0.). "ilius, we have
eight possible states for the three quarks: (t t t), (t t �), (t ./. t), (t � �), (� t t),
a t ./.), a � t) , and (l � l). But these are not the most convenient configurations
to work with, because they are not eigenstates of the total angular momentum. As
we found in Example 4.2, the quark spins can combine to give a total of or !, and
�,
f
1 181
1821 5 Bound St(]tel
L
8
A
3
L'
2
Fig. 5.12 OIt:>it�1 �ngular momenta for � three·body system.
L is the angular momentum of 1 and 2 ..bout their center of
m�ss (A); L' is the angular momentum of this combination
and 3 about the center of m..ss of all three (B).
the latter can be achieved in two distinct ways. Specifically,
IH) = (ttt)
IH) = (tt.!- + tH + Htl/./3
1 1 - i) = (Ht + 1.tJ. + tHl/./3
I�
-
(5.50)
spin 1 ('#"12)
(5.51)
spin � (,#"H)
(552)
� ) = (Hl)
IH)n = (t.!- - HI t /../2
11 ihl =t (t 1. - 1. t)l../2
I! �)H =J. (t.!- H)/../2
-
spin � (,#",)
-
�
I
I
The spin- combinations are completely symmetric. in the sense that interchanging
any two particles leaves the state untouched. The spino combinations are partially
antisymmetric - interchange oftwo particles switches the sign. The first set is anti­
symmetric in particles I and 2 - hence the subscript; the semnd is antisymmetric
n
i 2 and 3. We could also. of course. construct a pair of states antisymmetric in 1
and3:
I! lIn = Itt J. - J.tt)/../2
11 - 1)1l = (tH - Htl/../2
I
!
spin
! ("vtJ)
(553)
5.6 8Qryom
However, these are not independent ofthe other two; as you can check for yourself,
I }1l = I )12 + I b
(5.54)
In the language of group theory, the direct product of three fundamental
(two-dimensional) representations of S U(2) decomposes into the direct sum of a
four-dimensional representation and two two-dimensional representations;'
(5.55)
2 0 2 0 2 = 4 Gl 2 G1 2
A second respect in which baryons are more complicated than mesons has to do
with the Pauli exclusion principle. In its original formulation the Pauli principle
stated that no two electrons can occupy the same quantum state. It was designed to
state, 1/1 100 (there wouldn't be much left of chemistry if they did): they cannot,
explain why all the electrons in an atom don't simply cascade down to the ground
because the ground state can only accommodate two of them - one spin up, one
spin down. Once those positions are occupied, the next electrons are stuck in the
first excited state, n = 2, . . . , and so on. In this fonn, the Pauli principle seems
a little ad hoc, but it is actually based on something far deeper: if two particles
are absolutely identical, then the wave function should treat them on an equal
footing. If someone secretly interchanges them, the physical state should not be
altered. You might conclude from this that 1/1(1, 2) = 1/1(2, 1), but that's a little
too strong. Physical quantities are determined by the square of the wave function,
so all we can say for sure is that 1/1(1, 2) = ± 1/1 (2, 1): the wave function must
either be even - symmetric - or odd - antisymmetric - under the interchange of
two identical particles.t But which is it, even or odd? Nonrelativistic quantum
mechanics offers no answer; there are simply two classes of particles - bOOMS, for
which the wave function is even, andflrmions, for which it is odd. It is an empirical
fact that aU particles ofinteger spin are bosons, whereas those of t ·integer spin are
fermions. One ofthe major achievements ofquantum field theory was the rigorous
proof of this connection between 'spin and statistics'.
Boson (integer spin) => symmetric wave function : 1/1(1,2)
=
1/1(2, 1)
Fermion H-integer spin) => antisymmetric wave function : 1/1(1,2)
= -1/1(2, 1)
Suppose we have two particles, one in state 1/1", and the other in state 1/1�. If the
particles are distinct (one a muon and one an electron, say) then it makes sense to
• If the representations �re l�heled by spin. inste�d of dimensio�lity. Equ�lion s.ss re�ds ! ®
t ® l = i Qj j e l· Incidentally. it is a{jo possible to construct � spin. ! combination that is
.ymmtlri< in »>rticles 1 �nd 2: I ) _ I )ll + I )lJ. Somr �uthors prefer to use 1 h, �nd I ), in·
stead of I )tl �nd I hl.
t From It(l, 2)1' : It(2, 1)1' it follows only that �(I. 2) = �t,2. I). However. �pplyin8 the in·
terchange twiu brings us back to where we started. so ;;. = I . ..nd hen� � = ± I.
1 183
13(
I 5 Bound Sl(lt�s
askwhich s
i in state 1/1" andwhich is n
i state 1/1Ii. Thewave function forthe system is
if particle 1 is in 1/1� and 2 is in 1/IfJ' or
if it's the other way around. But if the two particles are indistinguishable, we cannot
say which one is in which state. If the particles are identical bosons, the wave
function s
i the symmetric combination
';(1,2) = (1/../2) [.;� (1)';fJ(2) + ';fJ(l)v" (2)]
(5.56)
and if they are identical fermions, the wave function is the antisymmetric combi­
nation
(5.57)
In particular, ifyou try to put two fermions (electrons, say) into the same state (V"
VIl) you get zero; il ean'tbe done. That'stheoriginal Pauli exclusion principle; but we
see now that it is not an ad hoc assumption, but rather a consequence ofa stroctural
=
requirement on the wave functions ofidentical particles. Notice, by the way, that the
Pauli principle does not apply to bosons; you can put as many pions into the same
state as you like. Nor s
i there any symmetryrequirement for distinguishab/t: particles;
that's why we didn't have to worry about it when we were constructing meson wave
functions (since one constituent is a quark and the other an antiql«lrk, they're alW(lYS
distinguishable). But in the case ofthe baryons we're putting three quarks together,
and this time we must take the antisymmetrization requirement into account.
Now, the wave function of a baryon consists of several pieces; there is the spatial
part, describing the locations ofthe three quarks; there s
i the spin part, representing
their spins; there is aflavor component, indicating what combination of u. d, and s
is involved; and there s
i a color term, specifying the colors ofthe quarks:
1/1
=
1/I (space) 1/I (spin)1/I(fiavor)v{color)
(5.58)
It is the whok W<lrks that must be antisymmetric under the interchange of any
two quarks: We do not know the functional form of the spatial ground-state
wave function, but it is surely symmetric; since I
I'
0, there is no angular
dependence at all. The spin state can either be completely symmetric U =
or of
mixed symmetry U =
As for flavor, there are 31 = 27 possibilities: uuu, uud,
udu, udd, . . , SS$, which we reshuffle into symmetric, antisymmetric, and mixed
combinations; they form r
i re<lucible representations of SU(3), just as the analogous
=
.
•
l).
=
!)
Notice t1ut • subtl� exknsion of the notion of 'id�ntiGl.l p.a.rtick· has implicitly � maok h�re,
for we a� tee.tins all quarks. regardless of color Or evm flavor. as different Slain of. single
I"'-rtid� 122].
5.6 8Qry<lns
spin combinations form
representations of SU(2). These are conYenientlydispiayed
in Eightfold-Way patterns:
"'"
(ddu + dud + utid 1/.,fJ
(dds + dsd +stidl/../3
(uud ... udu + OOu 1/.,fJ
(uds -t usd + dus + dsu + sud + soo 11.J6
•
hiss + sds + ssdl/,j'J
luus + usu + suulf..jJ
(USI + suS + 55ul/../3
.".: Completely tymrr>etric ltat...
•
Iuds -UId +<lsu -dU$ + sud - sdu 1/.J6
(ud-dulul..;2
[(UI -IU I d + (dl -wlu J 12
•
•
(us-suluA,/2
[2(ud-duls + (us - w l d - (m -Jdlu]/./i2
(us -$1.115/.;2
.J-, 2: Antlsymmetri(: in 1 �nd 2
1 185
136 1
5 Bound SIlIIt!
u (ud -dulIP
(dlus - w l +u(ds -.dll!2
d(ds -sd)l.J2
•
•
(2.(ud -du) + d{us -w ) -u (d. -sdl j/y'l"1
S(dS -sd)A/f
u(us-w)/y'2"
.(u.-w)/�
11-13' Amisvmmetrlc in 2 and 3
Thus the combination of three light-quark flavors yields a de(Uplet, a singlet, and
twooctets;' in the language ofgroup theory, the direo;:t product ofthree fundamental
representations of SU(3) deo;:omposes according to the rule
3 081 3 081 3 = 10 $ 8 $ 8 $ 1
(5.59)
Incidentally, we can also construct an octet that is antisymmetric in 1 and 3, but
this is not independent (Vru = 0/12 + 0/23); we have already used up the 27 states
avai
l able in making the four representations to, 8, 8, and 1.
(uud -duu )/,;2
(ulh-sdu + dus-wdI/2
(dds - Wd)/V'I
•
•
[2(usd -dw) + ud. -sdu -dus +wd! 1y'l"1
(uus -wu)/../2
(du -ud)/.j2
"'I J ; Amisvmmetric in 1 and 3
• As always in oclci (and nonet) diagrams. ! put rhe isotriplet ('I:o,) abow:, and the iso.inglet(s)
("A·) bene�th it. in the center.
5.6 8uty<>/IS
Finally, there is the question of color. In Chapter 1. I stated a general rule that
all naturally occurring particles are colorless; if a meson contains a red quark, it
must also contain an I2nlired quark, and every baryon must harbor one quark of
each color. Actually, this is a naive formulation ofa deeper law:
Every naturally occurring particle is a color singlet.
The three coum generate a color SUP) symmetry, just as the three light--quark
flavors generate flavor SU(3) . (The former is, however, an exact symmetry - quarks
ofdifferent colors all weigh the same - whereas the latter is only approximate.) By
putting together three colors, we obtain a color decuplet, two color octets, and a
color singlet (simply make the flavor --+ color transcription, u red, d --+ green.!
blue, in the diagrams above). But nature chooses the Singlet, and so for baryons
the color state is always
t(color) = (rgb - rbg + gbr - gtb + brg - bgr)/.../6
(5.60)
--+
--+
Because the color wave function is the same for aU baryons, we generally do not
bother to include it. However, it is absolutely crucial to remember that "1/r(color) is
antisymmetric, for this means the rest of the wave function must be symmetric. In
particular. in the ground state, with "1/r(space) symmetric, the product of t(spin)
and "1/r(flavor) has tobe completely symmetric. Suppose we start with the symmetric
spin configuration; this must go with the symmetric flavor state, and we obtain the
spin-� baryon decuplet:
t(baryon decuplet) = "1/r,(spin)t,(fiavor)
(5.61)
Write down the wave function for the 6.+, in the spin state mj "" - !
(never mind the space and color parts).
Example 5. 1
Solution:
1.6.+ :
� - i) = {(uud+udu+duu)/JJ}[HH + H� + tH)/..J3]
=
[uU)u(J.)d(t) + u(J.)u(tld(J.) + u(t)uU)d(l)
+ u(J.)d(J.)u(t) + u(�)d(t)u(�) + u(t)d(�)u(�)
+ d(�)u(J.)u(t) + d(�)u(t)u(�) + d(t)u(�)uu)}f3
For instance, ifyou could pull such a particle apart, the probability is � thatthefirst
quark would be a d with spin up, and * that it would be a u with spin down. _
The baryon octet is a little trickier, for here we must put together states of
mixed symmetry to make a completely symmetric combination. Notice first that
1 111
1&8 1
5 Bound $Iok.
the product of two antisymmetric functions is itself symmetric. Thus "'i2(spin) x
'" l2 (flavor) is symmetric in 1 and 2. for we pick up two minus signs when 1 ..... 2.
Likewise. "'n(spin) x "'2l(flavor) is symmetric in 2 and 3, and "'ll(Spin) )(
\l'o(flavor) is symmetric in 1 and 3. Ifwe now add these, the result will clearly be
symmetric in all three (for the normalization factor, see Problem 5.16):
\I'(baryon octet) == (../2f3H"'12 (spin)"'ll(llavOr)
+ \l'z)(spin)\I'zl(flavor) + \l'l)(spin)\I'u(ilavor)]
Exomple 5.2
Write down the spin/flavor wave function for a proton with spin up.
Soluticn:
Ip ; ! ! >
(5.62)
:::0
Ilft � t - � ftHudu - duu) + t(ft i - t � tJ(uud - udu)
J2
+t(tU - lft){uud - duujf ""3 == /uud(2 tU - U t - Ht)
+ udu(2 t H - Ht - tt l) + duu(2 Ht - t H - tU11
==
2
3v'2
I
'"
lv2
I
(u(t)u( t)dW) - ../2(u(t)ujl)d(t))
3
- l",(u(i)u(t)d(t)) + permutations.
3,2
_
If nothing else, I hope you will have gathered from this exercise that the
construction ofbaryon wave functions is a nontrivial business, in the quark model.
Apart altogether from the spatial wave function, there are three spins to juggle, as
well as three flavors and three colors, and it all has to be put together n
i a way that is
consistent with the Pauli principle. Perhaps, also you will forgive me for deferring
the explanation of how three quarks can generate the baryon octet (the decuplet,
remember, we got by naive quark counting back in Chapter 1). The essential point
is that the corners of the de<:uplet contain three identical quarks (1'1'1', ddd, and
sss); they necessarily form a symmetric flavor state, and hence must go with the
symmetric spin state U == �l. With two identical quarks (uud. say) there are three
arrangements (uud, udu, duu): you can make a symmetric linear combination,
which goes into the decuplet, and two of mixed symmetry, which belong to
SU(3) octets. Finally, with all three different, uds, there are six possibilities - the
completely symmetric linear combination completes the dewplet, the completely
antisymmetric combination makes an SU(3) singlet, and the remaining four go
into the two octets. Notice again the essential (if hidden) role of color in all this.
Without it we would be looking for antisymmetric spin/flavor wave functions;
spin � (symmetric) would have to go with the flavor singlet (antisymmetric). It is
possible to make a spino! octet without color (see Problem 5.18), but in place of
the decuplet we would have just one spin-� baryon. It was to avoid that disaster,
without sacrificing the Pauli principle, that color was introduced in the first place.
5.6 Baryons
6
5 .2
.
Magnetic Moments
As an application of the baryon spin/flavor wave functions, we now calculate the
magnetic dipole moments of the particles in the octet. In the absence of orbital
motion, the net magnetic moment of a baryon is simply the v«tor sum of the
moments ofthe three constituent quarks:
(5.63)
It depends on the quark flavors (because the three flavors carry different magnetic
moments) and on the spin configuration because that detennines the relative
(
orientations of the three dipoles). Apart from minute radiative corrections, the
magnetic dipole moment of a spn
i
point particle of charge q and mass m is
(Equation 5.18):
.
�
(5.64)
Its magnitude is
(5.65)
Sz
More precisely, this s
i the value of /-Lz in the spin·up state, for which
= hj2. It
is customary to refer to J.I., rather than p. itself, as 'the magnetic moment' of the
particle. For the quarks,
'''J.I..= ---3' 2m,c
(5.66)
Tbe magnetic moment ofbaryon B. then, is
J.l.B
= t 1(J.l.1 J.l.2 MI.IB t) 2 L' (B t l(J.l.iSi.IIB t)
Example 5.3
(B
+
+
=�
�,
(5.67)
Calculate the magnetic moment ofthe proton.
Solutian: The wave function was found in Example 5.2. The first term is
2
3../2 [U(t)u(t)dH))
Now
(/-LI St, + J.l.2S2. +
J.l.J SJ,)lu(t)u(t)dU))
= [J.I.�� J.I.�� J.l.d (-n] lu(t)u(t)d(�))
+
+
1 139
190
I S BOlmd Slares
Table 5.S M�gntlic dipolt momtnts of octtl baryons
Prediction
B�ryon
Momtnt
,
(illt" - (tljt�
(tlltd - (tIlt.
P
A
,
-
"'
"
"
SO
S-
.,
(!II-'" - W",
(�)(It" + I-'d) - (III-',
(jll-'d - Wit,
(jIlt, - (tIlt.
(jllt, -(}I"�
.
.
Experimtnt
-0.58
-0.613
2.68
2.458
2 79
2.793
- 1 86
-1.913
0.82
-LOS
-1.160
-1.40
-1.2S0
-0.47
-0.651
magneton. '''/2..." . S<>ura: Panicle Phyncs Bookl.t (2006).
Th� num�rical valu� ar� given as multiples ofthe nuclear
so this term contributes an amount
'
( 2 )' . �)U(t)u(t)d(�)I(It;S;
,JIU(t)u(t)d(�)) �(2J.tu
1:
r:;
3",2
;..!
=
- /-td)
Similarly, the se<ond term (u(t)u(j.)d(t)) gives fAltd, as does the third: We could
continue in this way to evaluate all nine terms, but the rest are simply permutations,
in which d occupies position 2 or position 1. The result, then, s
i
/-tu, /-td, and /-t, (Problem 5.19). The results are listed in the second column of
In this way we can calculate all the octet magnetic moments in terms of
Table 5.5. To getthe actual numbtrs, we need to knowthe quark magnetic moments
(Equation 5.66). Using the constituent quark masses m�
md = 336 MeVjc2,
m, = 538 MeVjc1, we obtain the figures in the third column of Table 5.5. The
comparison with experiment is reasonably good. considering the uncertainties in
the quark masses. Somewhat better predictions are obtaine<! f
i we take ratios. In
particular, to the extent that m. = md, we have
=
- = -.,
.,
2
3
(5.68)
which compares well with the experimental value, -0.68497945 ± 0.0000500 8.
•
Note that everything s
i normalized. so that for instance (w{f)lu(fll _ 1, and the $\at� � or·
thogonal (u(f)lu(l)
.. n.
..
5.6 Baryons
5.6.3
Masses
Finally, we turn to the problem of baryon masses. The situation is the same as for
the mesons: f
i flavor 5U(3) were a perfect symmetry, all the octet baryons would
weigh the same. But they don't. We attribute this in the first instance to the fact
that the 5 quark is more massive than u and d. But that can't be the whole story,
or the /I. would have the same mass as the "E's, and the 6.'s would match the
proton. Evidently, there is a sg
i nificant spin-spin ('hyperfine') contribution, which,
as before, we take to be proportional to the dot product of the spins and n
i versely
proportional to the product of the masses. The only difference is that this time
there are thru pairs of spins to contend with:
. 51 51 .51 +-5 . 51]
(5.69)
M{baryon) =m1 +m2 +m1+A, [51-mjm1 m2lml
mjm2 + -Here, A' (like A in Equation 5.46) is a constant, which we adjust to obtain the
optimal fit to the data.
The spin products are easiest when the three quark masses are equal, for
and hence
"
= -UU
+ I) - �I
2
for j
for j
= � (decuplet) )
= ! (octet)
(5.71)
Thus the nucleon (neutron or proton) mass is
] h'
M/"/ = 3m" - --A'
4 m;
the !J. is
Mt,
=
3m.. +
=
3m, +
and the n- is
Mo
(5.72)
] h'
--A'
4 m�
(S'?3)
I n'
--A'
4m:
(5.74)
In the case of the decuplet the spins are all 'parallel' (every pair combines to make
spin 1) so
(5.75)
1 191
192
1
5 Bound Slalt<
(and the same for 1 and 3, or 2 and 3). Hence for the decuplet
(5.76)
(which s
i consistent, notice, with Equation 5.71), and therefore
(s.n)
while
"
4
Me.' = Inw + 2m, + -A'
(-- I )
2
Inwln,
+ -,
In,
(5.78)
The 1: and A can be done by noting that the up and down quarks combine to
make isospin 1 and 0, respectively. and n
i order for the spin/flavor wave function to
be symmetric, under the n
i terchange of u and d. the spins must therefore combine
to a total oft and 0, respectively. For the 1:'s, then
(5.79)
whereas for the /I.
(5.80)
Using these results together with Equation 5.71, we find
"
= 2m... + m, + -A'
4
4)
C -z-
m.
m.m,
T�bIe 5.6 B�ryon octet �nd decuplet m�sses. (MeV/ell
B�ryon
N
A
E
S
•
E'
::;:*
"
COlkul.ued
Observed
939
939
1116
1114
1179
1327
1193
1318
1239
1381
1232
1385
>6"
1533
1672
1529
(5.81)
5.6 8aryon'
,nd
(5.82)
['U letyou figure outthe milSS of the 8's (Problem 5.22):
"
ME = 2m, + m. + -A'
4
( --)
=
1
'
m,
mu m,
2 -
(5.83)
Using the constituent quark masses m. = md 363 MeVj,Z, m. = 538 MeVjc},and
picking A' = (2m./fif SO MeV/,l, we obtain an excellent fit to the experimental
data (fable 5.6).'
References
1 R�ad�rs unfamiliar with quantum
medlanics will need to consult
an appropriate ttltbook for back­
euLalians pr�nt� Mre. S«. for
R. C. (1947) Physical Rowilw. n, 241.
6 Each normal mode of the electro­
ground. and for details of the ul·
Imlgnetic field functions as an os·
instance, hrk, D. (1992) Inlroduc·
grOWld·s�te energy of a harmonic
Ii"" to !he QUail/I'm Thro'1'. 3rd
til«>. S� Bjorken. ,. o. and Orell.
s. O. (1%4) Reialivutic Quantum Me·
elin, McGraw·HiIl, New York; (a)
Townsend. J. S. (2000) A Mode", Ap.
proach I<l QIulntum Mtchanico, Uni­
versity Science Books. Sau�ito. CA;
(b) Griffiths, D. /. (2005) Introduc­
tion to Quantum M�nicl. 2nd wn.
Prentice H;ill, Upper Saddle River,
N,.
2 For a fasdmoting account of the
experimental study of the hydnr
gen spectrum. from its beginnings
cillator; in qua.ntum mechanics the
oscillator is not zero. but rather.
chanics. McGraw-Hill. New York.
p. S8.
7 Sethe. H. A. and Salpeter, E. E.
(1977) Quantum MWlanics O/ OM'
and Two·EUctron Atoms, Plenum.
New York. Sect. 21. for details,
8 Bethe. H. A. and Salpeter, €. E.
(1977) Quantum Muhanics of
One- and Two·E/edron AUlms,
to th� pres�nt o:by. see th� am·
9 For an accessible account of hyper·
A. L. �nd Series. G. W. (March
hydrogen. s� Griffiths, O. J, (1982)
S�, for example, Griffiths, D. J.
'98.
10 Ewen. H, l. and Purcell. E. M. (1951)
Nllture. 168. 356.
II The spectrum of positronium is dis·
in th� mid·ninet�nth century up
cl� by H�nsch. T. w.. Sch.awlow.
1
5 L:;omb. W, E. Jr. and Retherford.
1979) Scientific American.
')4.
(1999) Introduclion 10 E/«Irodynam.
ics. 3rd edn, Pr�ntlce Hall. Up�
S�ddle River. NJ; Problem 5.5(;.
4 For the calcubtion, S� Kinoshita,
T. and Nio. M. (2006) Ph�ical Re­
view D. 73. 013003; for the mea·
surement. S� (a) Odom. B. tt aI.
(2006) Ph}'3iw/ Rowi.... Ulltrs. 97.
030801.
Plenum. New Vork. p. 110.
fin� splitting in the ground state of
Amtrican Journal of Physics, 5<1.
cussed in Bethe. H. A. and Salpeter.
E. E. (1977) Quanlum Mr.dwnics
of OM' and Two·Eltclron Atoms.
Plenum, New York; Sect. 21. for de·
tails. S«t. 23. S� also (a) Berko. S.
and Pendleton. H. N. (1980) Annual
• Noti�. h�r. that _ are obliged to use somewhat diffi=1 constituent quark I1141Sse5 in
Tables 5.3, 55, and 5.6, as I "",mN you in the footnote to Table 4.4.
1 193
19-41 5
Bound Stales
Rtvitw ofNuckar and Partick Sci·
18 The 'MIT Bag Model' offers a
enet, 30, 543: and (b) Rich, A. (1981)
possible approach to relativistic
light-quark systems. but �t the cost
of vastly oversimplified dynamics.
The qUllrks are tre�ted �s rr� par·
ticles confined within � spherical
'b�g', which is stabilized by an ad
hoc external pressure. Many in­
teresting uIruiations have been
carried out using the bag model.
but no one would pret",nd tlu.t it is
a realistic picture of h.;odron struc­
ture. S� Close, F. E. (1979) An In­
lroducfwn 10 QIUlri<s and Pano".,
Academic, London; Chapter 18.
n See, for e�mple, Halzen, F. and
Martin. A. D. (1984) Quarks and up­
ton!, lohn Wiley & Sons, N� York,
p. 42.
2(1 (20%) &,,;ew of Panicle Phl"ics;
Section 14.2.
21 Gasiorowicz, S. and Rosner, I. L
&vi=<; of Modtrn Physics, 53, 127.
12 The details of w fin", structur",
�� not quit'" W S;tme in qu�rko·
nlum �s n
i positronium; � Ekh�n,
E. and Feinberg. F. (1979) PhY'"
;':(1/ Review !.tUell, 43. 1205; �nd
Phys;':QI Rel'icw. (1981) 023, 2724.
13 There are other ways to esti·
mate Fo, which gi"'" roughly the
sam", �nswer. S� Perldns, D. H.
(2000) Inlroduction to HJgh En­
ergy PhY"ics, 4th edn, Cambridge
Uni�rsity ?ress, Cambridge, UK;
Section 6.3.
14 Appelquist, T. and Politzer, H. D.
(1975) PhY"ical Rev;';'" l.dtell, 34, 43;
(1975) Physical Review, 012. \4(14.
IS For �n interesting account of
wse discoveries. see the �r·
tiele 'Quarkonium' by Bloom.
E, D. �nd Fe!dm�n, G. ), (May
(1981) American JournQl ofPhY"ics.
49, 954; an article that contains
1982) S,�nfific American, 66.
a wealth of useful and accessibl", infonnation about the quark
modeL
22 Pais. A. (1986) Inward Bound. Ox·
ford University Press, Oxford,
UK. p. 425; attributes the e��n­
sion to (a) Cassen, B. and Condon,
E. U. (1936) Pkysical Rev;';"', SO.
16 Eich�n. E. and Gottfried. K.
(1977) Physics !.ttlm B. 66, 286.
17 The effect of vacuum polari:ution
on the spectrum of bottomonium
is explored in an accessible m·
el", by Conway, ). el QI. (2001) Eu­
ropean journal of Phy,iokigy, 22,
8<6.
533.
Problems
5.1 (I) Thedeuteron's mass is 1875.6 MeV/c1. Whatis its binding energy? Is this a relativistic
system?
(b) Ifyou take the lip- �nd down-quark m�sses 10 be those given in T�ble 4.4. wh�t is w
binding energy ofa pion? Is this a relativistic system?
5.2 Use Equation 5.12 to obtain the ground·stat'" wave function '/1100. Show tlu.t it satisfies
the SdtrMinger eqUlltion (EqUlltion 5.1). with the appropriate energy. and ched: thitt it
is properly nonnalized. IAnswer: "'100 (lj.J'if,il)e-'/'e-IEl'lhl
5.1 Work out all ofthe hydrogen w;o.ve fu�tions for n = 2. using Equation S.12. (How many
are th",re?)
5.4 Using Equation 3,43 to express the kinetic energy (T E mc'l) in terms of p (and mI.
show tlu.t the lowest-order relativistic co,.,...,.;tion to T pI/2m is _p'/8m3cl.
S.S Find the energy splitting between the) ! andj j levels for n 2 (Figure 5.2). in
electron volts. How does this compare with the spacing belW�n the " 2 and n I
Bohr ",nergi"'S?
5.6 Estimate the Lamb shift energy gap between the 2SI/l and 2PI/l levels in hydrogen,
using Equations 5.20 and 5.21. What is the frequency of the photon emitted in such �
transition? (The experimental Villue is 1057 MHz.)
'"
=
-
=
=
=
=
=
""
5.6 Bory,,",
5.1 Ifyou indudethe hnestrocture, Loomb shift, and hyperhne splitting, how m�nydifferent
n = 2 energy l�ls are there altogether in hydrogen? Find the hyperfine splitting betwef:n
the lSLll and 2PLp levels, and compare the Loomb shift (Problem 5.6).
5.8 Analyze the splitting ofthe 11 = 3 Bohr !-eve! n
i positronium. How many different levt:ls
are there, and what are their relativt: energies? Construct the level diagr.lm, analogous
to Figure 5.6
5.9 Would you consider the ,pIsS} meson bound or qUl1S;·bound?
5.10 On dimensional grounds, show th,u the energy levels of a purely linear potential. VIr)
= For. must be ofthe form
"
E =
((F�)2)I/l
tl"
(5.84)
where a" is a dimensionless numerical factor.
5.11 Use the numerical results in Table 5.2 10 'predict' the masses of the four lighiesl <It's
and T"s; compare the experimental results (Figure 5.9). What value ofFo givt:s the besl
fil to the l�l spacings? Why aren'l the Q]culated masses n
i better agreement with the
experiments?
5.12 Using Equation 5.46, with the values of m., md, m" and A given in the text, Q]culate
the meson masses in T�b1e 5.3. [Hint: For the 'I, first find the mass for pure uii, pure
ad, and pure ';, and think of the 'I as being �uii, �dd. and }';.J Also apply the formula
'
'
to the I) . �nd note the dis;lstrous result. [For commentary on the II mass problem, see
Quigg. C. (1983) Gauge TheorUs oftlu Strong, Weak, and Elearomognaic Inleraaion"
Benjamin, New York, p. 252.J
5.13 In the text, we ustd Equation 5.46 to Q]culate the masses of light·quark pseudoscalar
and vector mesons, BUI the s;lme formula can be applied to heavy-quark systems
n
i volving ch.arm and beauty quarks.
(a) Calculate the masses of the pseudoscaln mesons 'l,{cC), I)il(eiij, Dt(cS'), and the
corresponding ve.::tor mesons '" (cC). !),"o (eii), and D'+(ei). Compare theexperimental
values, from the Partide Data Booklet.
(b) Do the same for the 'bottom' mesons ub, sb, cb, and bb. At present only the
pseudoscalars B+(ub). if,'(sb). Bi(cb) and the ve<:tor l' (bb) h.avt: been detected
experimentally.
5.14 Construct the eight states >JI'l in Section S.6.1. [Hint; The six outer ones are e�sy - the
qu�rk content is determined by Q and S, and all you h.ave to do is antisymmetrize
in 1 and 2, To get the two states in the center, remember that the one in the 'Eo,
position forms �n isotriplet with the 'E+' �nd '1:: -'; the 'A' Gin then be constructed by
orthogonali1.ing with respect to ':1:0' and t.1.1
�.15 Construct the (singlet) color wave function for mesons, analogous to Equation S.W,
5.16 Check that the baryon octet spin/flavor wave function (Equation 5.(0) is correctly
normalized. Remember that <ltLl is not independent of t'l �nd >/tn.
U7 Construct the spin-flavor w�ve functions. as in Example S.2, for E+ with spin up and
A with spin down.
5.18 Construct a totally anlisymmetric spin/flavor h;.ryon octet. (!n this configuration we
do not need color to antisymmetrize the wave function. However, an antisymmetric
decuplet �annot be constructed. See Halzen and Martin, Reference 1!9), Exercise 2.18.)
5.19 (a) Derive the expressions in the second column ofTable 5.5.
(h) From these results, calculate the numbers in the third column ofTable 5.5, using the
quark masses given in the text.
5.20 Calculate the r�tio /-1."//-1., in the configuration you found for Problem 5.18 Notice that
1-<, is ntgalive in this ase! Is your result consistent with experiment? (Here, then, is a
second strike against the quark model without color, the first strike being its failure to
account for the dKuplet.)
1 195
1961 S Bound Stol,s
s.u Showth;..tjJ.p+ '" -I-'p- = 1-',. (Se.:o H�lun and Martin. R�f�I�nc", 119), ERlci� 2.19). As
far u J know, th", magn�tk dipol� moments of�tor m",sons ha� not �n m",�sured.
5.22 US'" Equation 5.69 to d"'teTmin", th", m�ss ofth'" 8.
5.2.3 Using Equations S.12. S.B. and S.28, calcuL:u", th", ",lectrond",nsity�t th", location ofth",
positron, in th",ground state ofpositronium, 11/1100(0)11.
6
The Feynman Calculus
In this chapter, wt: begin the quantitativeformulation ojdtmtnUiry particle dynamics,
which amounts, in practiu, to tlit w!culation of decay ralts (f) and scattering cross
sections (u). 'Tht procedure invclvt:s two distinct paris: (1) t:Va/U(Jtion of the rekvant
Ftynman dwgnllns to determine llit 'amplitude' (.4) for the process in question and
(2) insa/ion of.4t into Fermi's 'Colden Ruk' to computt r oru, as the cast may be. To
Ihtorns
QED. QeD, and cwS - art developed in succuding chapters. [fyou lilu,
avoid distracting algebraic complications, r introduce hert a simplifid model Realistic
-
Chapter 6 can be rt(.Id immediately after Chapter
3. Study it with scrupulous can, or
whatfoUows will be unintelligibk.
6.1
De<:ays and Scattering
As I mentioned n
i the Introduction, we have three experimental probes of
elementary particle interactions: bound states, decays, and scattering. Nonrela­
tivistic quantum mechanics (in Schrodinger's formulation) is particularly well
adapted to handle bound states, which is why we used it, as far as possible,
in Chapter S. By contrast, the relativistic theory (in Feynman's formulation) is
especially well suited to describe decays and scattering. In this chapter I'll n
i ­
troduce the basic ideas and strategies of the Feynman 'calculus'; in subsequent
chapters we wi
l l use it to develop the theories of strong. electromagnetic, and weak
interactions.
6.1.1
Decay Rates
To begin with. we must decide what physical quantities we would like to calculate.
In the case of decays, the item of greatest n
i terest is the lifetime of the particle
in question. What precisely do we mean by the lifetime of, say, the muon? We
have in mind, of course, a muon at rest; a moving muon lasts longer (from our
perspective) because of time dilation. But even stationary muons don't all last the
same amount of time, for there is an intrinsically random element in the decay
Inlrod..aion. 10 /:1e",,,"WI}' l'articks. SecotuI &lilio�. D�vid Griffiths
Copyrighle 2008 W[LEY·VCH VefL>g GmbH & Co. KG.A, Weinheim
ISBN: 978+527-40601-2
1931 6 The Feynman Calculus
process. We cannot hope to calculate the lifetime of any particular muon; rather.
what we are after s
i the average (or 'mean') lifetime, f, of the muons in any large
sample.
Now, elementary particles have no memories, so the probability of a given muon
decaying in the next microsecond is independent of how long ago that muon was
created. (It's quite different in biological systems: an 80-year-<lld man is much more
likely to die in the next year than s
i a 20.year-old, and his body shows the signs
of eight decades of wear and tear. But all muons are identical, regardless of when
theywere produced; from an actuarial point of view they're all on an equal fooling.)
The critical parameter. then, is the duay rate, r, the probability per unit time that
any given muon will disintegrate. Ifwe had a large collection of muons, say, N(t),
at time t, then Nrdt of them would decay in the next instant dt. This would, of
course, durtase the number remaining:
dN = - rN dt
(6.1)
It follows that
N(t) = N(O)e-n
(6.2)
Evidently, the number of particles left decreases exponentially with time. As you
can check for yourself (Problem 6.1), the mean lifetime is simply the reciprocal of
the decay rate:
T
=I
(6.3)
r
Actually, mostparticles candecay by several different routes. The ;rr +, for instance,
usually decays to �+ + vU. but sometimes one goes to t+ + II,; occasionally, a ;rr +
decays to �+ + lIu + y, and they have even been known to go to t+ + II, + ;rro. In
such circumstances, the tetal decay rate is the sum of the individual decay rates:
•
(6.4)
i the redprocal of rto,:
and the lifetime ofthe particle s
(6.5)
In addition to T, we want to calculate the various branching ratios, that is, the
fraction of all particles of the given type that decay by each mode. Branching ratios
are determined by the decay rates:
Branching ratio for ith decay mode = r;/ r,ot
(6.6)
For decays, then, the essential problem is to calculate the decay rate ri for each
mode; from there it is an easy matter to obtain the if
l etime and branching ratios.
6.1.2
Cross Sections
How about scattering? What quantity should the experimentalist measure and
i calculate? If we were talking about an archu aiming at a 'bull's·eye',
the theorst
the parameter of interest would be the size of the targtt or, more precisely, the
cross·sectional area it presents to a stream of incoming arrows. In a crude sense,
the same goes for elementary particle scattering: f
i you fire a stream of electrons
into a tank ofhydrogen (which is essentially a collection of protons), the parameter
of interest is the size of the proton - the cross sectional area I]' it presents to the
incident beam. The situation is more complicated than in archery, however, for
several reasons. First of all the target is 'soft'; it's not a simple case of'hit·or·miss',
but rather 'the closer you come the greater the deflection'. Nevertheless, it is still
possible to define an 'effective' cross section: I'll show you how n
i a moment. Sec·
ondly, the cross section depends on the nature ofthe 'arrow' as well as the structure
ofthe 'target'. Electrons scatter off hydrogen more sharply than neutrinos and less
so than pions, because different interactions are involved. !t depends, too, on the
outgoing particles; Ifthe energy is high enough we can have not only dastic scattering
(t + p -4 t + pI, but also a variety of inelastic processes, such as t + p _ t + p + y,
or ( + p + ;rro. oreven, in principle, II, + A. Each oneofthese has its own ('exclusive')
scattering cross section, qi (for process i). In some experiments, however, the final
products are not examine<!, and we are interested only in the /-otal ('inclusive') cross
section:
·
(6.7)
Finally, each cross section typicallydepends on the vtlocity ofthe incident particle.
At the most naive level we mg
i ht expect the cross section to be proportional to the
amount of time the incident particle spends in the vicinity ofthe target, which s
i to
say that I]' should be inversely proportional to v . But this behavior s
i dramatically
altere<! in the neighborhood of a 'resonance' - a spedal energy at which the
particles involved 'like' to interact. forming a short·Jived semioound state before
breaking apart. Such 'bumps' in the graph ofq versus v (or, as it is more commonly
plotted, I]' versus E) are in fact the principal means by which short·lived particles
are discovered (see Figure 4.6). So, unlike the archer's target, there's a lot ofphysics
in an elementary particle cross section.
Let's go back, now, to the question of what we mean by a 'cross section' when
the target is soft. Suppose a particle (maybe an electron) comes along. encounters
some kind ofpotential (perhaps the Coulomb potential ofa stationary proton), and
scatters offat an angle O. This scatttringan&it' is a function of the impact paranuter b,
the distance by which the incident particle would have missed the scattering center,
had it continued on its original trajectory (Figure 6.1). Ordinarily, the smaller the
impact parameter, the larger the deflection, but the actual functional fonn of O(b)
depends on the particular potentia! involved.
,
,
,
__ _ _ _ _ _ _ _ _
b
J
,
,
,
,
,
"., .
�
_ _
_ _ _
- -- - -- - - - - -+- - - - - - -
Scattering center
Fig. 6.1 Suttering from � fu<!d potential: () is the sattering
angle ,lnd b is the ;m�ct parilmeter.
Example 6. 1 HQrd·sphere SCQltering Suppose the particle bounces elastically off a
sphere ofradius R. From Figure 6.2, we have
b = Rsina,
Thus,
sin O' = sin(1T/2 - B/2) = cos(0/2)
and hence
b = Rcos(B/2)
or (} = 2cos-1 (bIR)
This s
i the relation between (; and b for classical hard-sphere scattering.
b
Fig. 6.2 Hard-sphere scattering.
R
r::t!
6.1 DeCQYS Qnd Sronering
Fig. 6.3 Particle incident in area dO" scatters into solid angle dn.
i with an impact parameter between b and b + db, it will
If the particle comes n
emerge with a scattering angle between /:I and /:I + d8. More generally, if it passes
through an infinitesimal ana dO", it will scatter into a corresponding solid angle
dO (Figure 6 .3). Naturally, the larger we make do-, the larger dO will be. The
proportionality factor is called the diffirtntial (Scattfring) cross stetkm, D:*
(6.8)
do- = D(@) dQ
The name is poorly chosen; it's not a differential. or even a derivative, in the
mathematical sense. The words would apply more naturally to do than to do jdn .
but I'm afraid we're stuck with it.
Now, from Figure 6.3 we see that
dO" = lbdbd¢I,
dQ = l si n 8 d 8 d¢1
(6.9)
(Areas and solid angles are intrinsically positive, hence the absolute value signs.)
Accordingly,
do
1 b ( db )1
0(9) = dn = sin8
de
Exampl� 6.2
(6.10)
In the case of hard-sphere scattering, Example 6.1, we find
• In principle D an depend on the azimuwt angle r/>; however, most potoo:nti.11s of interest are
spherically symmetrical, i n which ase the differential cross section depends only on 9 (or. if
you prefer. on b). By the way, the notation (0) s
i my own; 1nO$! �ple just write doIdn, and i n
the rest o f the bool: I"ll do the same.
I Wl
2021 6 TM Feyllman Calculus
and hence
'in
�I'�/2") = W- COs(On) sin(O/2) R2
�
= R"�b�
2sinO
"'2 sinO = 4
D(O)
-
"'"
Finally, the /{)tal cross se<:tion is the n
i tegral of do- over all solid angles:
17 =
f f
Example 6.3
dl7 =
D(O) dQ
(6.11)
For hard.sphere scattering,
which is, of course, the total cross se<:tion the sphere presents to an incoming
beam: any particles within this area will scatter, and any outside will pass by
unaffe<:ted. i.E
As Example 6.3 indicates, the formalism developed here is consistent with our
naive sense of the term ·cross section', in the case of a 'hard' target; its virtue is
that it applies as well to 'soft' targets, which do not have sharp eclges.
Example 6.4 Rutherford Scattering A particle of charge ql scatters off a stationary
particle of charge qz. In classical me<:hanics, the formula relating the impact
parameter to the scattering angle is [1]
b
= q��2 cot(O/2)
where E is the initial kinetic energy of the incident charge. The differential cross
section is therefore
[n this case, the total cross section is actually infinite:'
17
1
.
.
= 2n (Q,Q')'1"
-•
smOdO
= oo "��
4E
sin (0/2)
0
Suppose we have a beam of ncoming
i
particles, with uniform luminosity C (C is
the number of particles passing down the line per unit time, per unit area). Then
• This is re4ted to the bct th�t the Coulomb �mi�1 has n
i finite ..n8" 1see footnote in
Stction 1.3).
;/)
6.2 The Golde.. R�le
"",�,o,
/Ko
£ ------t.� .
Incident beam
Target
Fig. 6.4 Sc�tte'ing of � beam with lumino�ity C.
dN = £ da is the number of particles per unit time passing through area dO", and
hence also the number per unit time scattered into solid angle dO:
dN = C dO" = £D(9) dO
(6.12)
Suppose I set up a detector that subtends a solid angle dO with respect to
the collision point (Figure 6.4). I count the number of particles per unit time
(dN) reaching my detector - what an experimentalist would call the tvtnl rate.
Equation 6.12 says that the event rate s
i equal to the luminosity times the differential
cross section times the solid angle. Whoever is operating the accelerator controls
the luminosity; whoever set up the detector determined the solid angle. With these
parameters established, the differential cross section can be measured by simply
counting the number of particles entering the detector:
da
dQ
dN
C dr.!
(6.13)
If the detector completely surrounds the target, then N = (fC; as accelerator
physicists like to say, 'the event rate is the cross section times the luminosity'.'
6.2
The Golden Rule
In Section 6.1 I introduced the physical quantities we need to calculate: decay rates
and cross sections. In both cases there are two ingredients n
i the recipe: (i) the
ampliludt (...Ii) for the process and (i
i) the phase space available.t The amplitude
contains all the dynamical information; we calculate it by evaluating the relevant
Feynman diagrams, using the Ftynrnan rults appropriate to the interaction n
i
in the identity of the p<.rticip<.nts during the
scattering process lin the rcaction ,.- + p.. .....
K+ + I:-. for eumple. dQ might represent
particle is simpl� Ikfluud as it passes through
the solid angt.: into which the K+ scatters).
the scattering potrnti�l My purpose was to
t The ..",plih.1k is also called the ",,,,rix tk·
introduce the essential ideas in the simplest
....111; the p"= '»<IU is sometimes called the
possible context. But in Se.:tion 6.2 the for·
Iknsity offinal $lam.
tmIism is completely general; it n
i cludes the
re.:oil of the torget, and �lIows for a ch<lnge
, In this discussion, I have assum� thlt the
tUg<'t itself s
i $lalio....1")' �nd th�t the incident
1 203
2041 G Tht Fey"ma" Calculus
question. The phase space factor is purely kirumlltic; it depends on the masses,
energies, and momenta ofthe participants, and reflects the fact that a given process
is more likely to occur the more 'room to maneuver' there is in the final state. For
example, the decay of a heavy particle into light secondaries involves a large phase
By contrast, the decay of the neutron (II ...... P + t + ii.), in which there is almost
space factor, for there are many different ways to apportion the available energy.
no extra mass to spare, is tightly constrained and the phase space factor s
i very
small:
The ritual for calculating reaction rates was dubbed the Go/dm Rule by Enrico
Fermi. In essence, Fermi's Golden Rule says that a transition rate s
i given by
the product of the phase space and the (absolute) square of the amplitude. You
may have encounteredthe nonrelativistic version, n
i the conteJrt oftime-dependent
field theory [3]. I can't dtriJlf! it here; what I will do is stllit the Golden Rule and try
perturbation theory [2J. We need the relativistic version,whichcomes fromquantum
to make it plausible. Actually, I'll do it twice: once in a form appropriate to decays
and again in a form suitable for scattering.
6.2.1
Golden Rule (or Decays
Suppose particle 1 (at rest)t decays n
i to several other particles 2, 3, 4, . . . , II:
1 ...... 2 + 3 + 4 + · · · + n
(6.14)
The decay rate is given by the formula
(6.15)
where m, is the mass of the ith particle and Pi s
i its four·momentum. S s
i a
statistical factor that corrects for double·counting when there are identical particles
in the final state: for each such group of 5 particles, S gets a factor of (I/s!). For
instance, if Il ...... b + b + c + c + c, then S = (1/2!)(1/3!) = 1/12. If there are 110
identical particles in the final state (the most common Circumstance), then S = 1.
Remember: The dynamics of the process is contained in the Ilmplitudt, ...t'(p"
P2, . . ,pn), which is a function of the various momenta: we'll calculate it (later)
•
For l morr �trrmr c;,s.,. consid�r lhr (kinrrmtially forbiddrn) dNay Q- """* ::;- + X'. Sincr
thr final products w,,;gh more than th� n, lh�r� is no phase s�ce avai!abl� at all ..nd lh� decay
t There is no loss of generality in assuming plrtidr 1 is al r...l: this is simply an aslUlr choice of
rate is zero.
""ferrnce fra,"".
6.Z The Golden Rule 1 205
by evaluating the appropriate Feynman diagrams. lbe rest is phase space; it tells us
to integrate over aU outgoingfour-momenta, subject to wee kinematical constraints:
J J
E1 - pJ c2 = mJ c�), lbis is enforced by the
delta function ;S (pJ - mJ c2) , which is zero unless its
1, Each outgoing particle lies on its mass shell: p = m cl
(which is to say,
argument vanishes:
J
2. Each outgoing energy s
i positive: p = Ej/c > 0, Hence the I)
function.t
3. Energy and momentum must be conserved: PI = PI + PJ .
+ Pn . lbis is ensured by the factor ;S4(p1 P2 Pl . , . -p�) .
- -
lbe Golden Rule (Equation 6.15) may look forbidding, but what it actually says
could hardly be simpler: all outcomes consistent with the three natural kinematic
constraints are a priori equally likely. To be sure, the dynamics (contained in Al)
may favor some combinations of momenta over others, but with that modulation
you just add up aU Iht possibilities. How about all those factors of21l ? These are easy
to keep track of ifyou adhere scrupulously to the following rule:*
Every ;S gets (21l); every d gets 1/(21l).
(6.16)
Four-dimensional 'volume' elements can be split into spatial and temporal parts:
(6.17)
(I'll drop the subscript j, for simplicity - this argument applies to each of the
outgoing momenta). The po integraIsS can be performed immediately, by exploiting
the delta function
(6.18)
Now
(6.19)
• Ifyou ar� unfamili�r with th� Dir.lc ddta function. you m ....t study Appendix A orefully before
proceeding.
t 9�x) is the jH�..yjside) $� function: 0 if " < 0 and 1 if " > 0 jseo: Ap�dix A).
t Some of
these factors eventu..lly onc�l out. and you might wonder if there $
i " mor� efficient
way to ma....� them. I don'l think SQ. F�man s
i suppo� to h.ov� shout� in ex:isl"'ration
(at a gradwote student who 'couldn't be boIher� with such trivial m..tters') 'If you on'l �t th�
br's right, you don't know nolhin.g!'
I The integral sign in Eqwo!ion 6.15 actually stands for 4(n - 1) int.:grations - on� for �a,h com­
ponent of the � - 1 outgoing momenta.
2061 6 The FeynmGn Calculus
(see Problem A.7), so
(the theta function kills the spike at pO = _Jpl + m2cl, and it's 1 at pO =
Jpl + m1c2). Thus Equation 6.15 reduces to
(6.21 )
with
(6.22)
wherever il appears (in ..$( and in the remaining delta function). This is a
more useful way to express the Golden Rule, though it obscures the physical
content:
6.2.1.1
Two-particle Decays
In particular, iflhere are only two particles in the final stale
(6.23)
The four-dimensional delta function is a product of temporal and spatial parts:
i at rest, so PI
But particle I s
replaced (Equation 6.22), sot
(6.24)
-
0 and p� = mlc. Meanwhile, p� and p� have been
(6.25)
• You might �ognizr tht quantity
JP] + mJe1 as Ej/e, �nd tmny books write it this wa�. It's
dangerous notation: Pj is an integration variable. so Ej is not so"", (On.tan! you can take oU!·
side thr integral. Usr i! as shorth.o.nd, if you like, but �membrr that
is a function of Pi' not
an indep<"ndent variable.
t We un drop the minus sign n
i the fimol dtlta function, siner 5(-x) '" 6(x).
Ej
6.2 The GoidM Rule
The Pl integral is now trivial: in view ofthe final delta function it simply makes the
replacement
Pl --I-
-P2
(6.26)
leaving
For the remaining integral we adopt spherical coordinates,
-jo
sin OdrdO d4' (this is
space, ofcourse: =
dJ p2
r2
--I-
P2 (r. e, 4'),
r IPll).
momentum
' (mlc - Jr2 + �c2 - Jr2 + �c2 )
J,z + m�clJr2 + mfc2
r� �
'=2 / I"'I' --'--r==�=c?e=='''7'--'S
3211" /iml
x
,1 sinO drdO dip
(6.28)
Pl' and Pl, but PI =
Now, Ai was originally a function of the four·momenta
0) s
i aconstant (as far as the n
i tegration is concerned), andtheinte rats already
Pl ,
(mlc,
performed have made the replacements p� -jo
p� -jo
+
+
and
-jo -P , so by now Ai depends only on P As we shall see, however, amplitudes
2
l
must be SC(l/ars, and the only scalar you can make out of a vector is the dot product
= ,2. At this stage, then, ..A is a function only of r (not of 0 or
with itself:'
4'). That being the case we can do the angular integrals
.
pJ
jp� m�c2.
P� m�c2 ,
P2 . P2
1"
sin O de =
2,
ll�
d¢ = 21T
(6.29)
r
and there remains only the n
i tegral:
To simplify the argument ofthe delta function, let
(6.31)
• If the p.a.rtidrs carry spin, thrn .At might drp�nd also on (PrSj\ and (Si·Sj). Ho�. sjn<e 0;.
periments rarely measure the spin orienution. we almost alw..ys work with t� spin·..veragw
amp�tudr. In that OISO'. and of Course in the case of spin O. the only vrctor in sight is PI and
the only scabr v..riab1e is (p,)l.
1 207
2081 6 Th� Ftrynmon CO!GI4!I"
'0
du
dr =
ur
Jr2 + miC1Jrl + m�c2
(6.32)
Th,n
(6.33)
The last n
i tegral sends' u to ml', and hence r to
(6.14)
i the particulur
(Problem 6.5). Remember that r was short for the variable IP21 ; ro s
value of Ip 1 that is consistent with conservation of energy, and Equation 6.25
2
simply reproduces the result we obtained back n
i Chapter 3 (Problem 3.19). In
more comprehensible notation, then,
r = �I
Afll
8nhmrc
(6.35)
where Ipi is the magnitude of either outgoing momentum, given n
i terms of the
three masses by Equation 6.34, and Af is evaluated at the momenta dictated by
the conservation laws. The various substitutions (Equations 6.22, 6.26, and 6.34)
have systematically enforced these conservation laws - hardly a surprise, since they
were built n
i to the Golden Rule.
The two-body decay formula (Equation 6.35) is surprisingly simple; we were
able to carry out all the integrals without ever knowing the jUnctional form of
Mathematically, there were just enough delta functions to cover all the variables;
physically, two-body decays are kinematically cktumined: the particles have to come
out back-to·back with opposite three-momenta - the direction of this axis is not
fixed, but since the initial state was symmetric, it doesn't matter. We wi
l l use
Equation 6.35 frequently. Unfortunately, when there are three or more particles in
the final state, the n
i tegrals cannot be done W11i
1 we know the specific functional
form of Af. In such cases (of which we shall encounter mercifully few), you have
to go back to the Golden Rule and work it out from scratch.
Af!
6.2.2
Golden Rule for Scattering
Suppose particles 1 and 2 collide, producing particles 3, 4, . .
.
, n:
1 + 2 -1- 3 + 4 + · · · + n
(6.36)
• This �ssumes m, "" (ml + m)); otherwise the dell<! function spike is outside the domain of
integution �nd we goet r '" Q. recording the f�ct tb�t � p.>rticie c;onnot deu.� into huvier
=nd�ries.
6.} The Golden Rille
The scattering cross section is given by the formula
d Pi
, n 2d (p' - m. 2) e (p.0) (211")4
•
,-,
•
,
J
'
J
J
'
--
(6.37)
where Pi is the four·momentum of particle i (mass mil and the statistical factor
(S) is the same as before (Equation 6.15). The phase space is essentially the same
as before: integrate over all outgoing momenta, subject to the three kinematical
constraints (every outgoing particle is on its mass shell, every outgoing energy is
positive, and energy and momentum are conserved), which are enforced by the
i , we can simplify matters by performing the
delta and theta functions. Once agan
P'l integrals:
(6.38)
with
(6.39)
wherever it occurs in .41 and the delta function.
6.2.2.1 Two-body Scattering in the eM Frame
Consider the process
1 + 2 -+ 3 + 4
(6.40)
in the CM frame, pz = -PI (Figure 6.5). where
(6.41)
(Problem 6.7). In this case, Equation 6.38 reduces to
(6.42)
1 209
210
I 6 The FeynmOln OJlwllls
_ . -.--
p,
'"
P4
"
.'-�--.
/ After
Before
Fig. 6.5 Two·body scattering in the eM frame.
As before, we begin by rewriting the delta function:'
(6.43)
i sert Equation 6.39 and carry out the P. integral (which sends P. -+
Next we n
-p}):
x
oS
[
(EI + E2)/e -
jp� + mte2 - jp� + m!c1]
(6.44)
This time, however, IAI2 depends on the directiol1 of p} as well as its magnitude,t
so we cannot carry out the angular integration. But that's all right - we didn't
really want (F in the first place; what we're after is dCF Idn, Adopting spherical
coordinates, as before,
dlpJ = r2 dr dn
(6.45)
(where r is shorthand for Ip}1 and dn = sine de d4'), we obtain
do
dO
=
( • )'
8rr
s,
(El + E1) lpd
r oo I "'I'
Jo ·
(6.46)
, Ob..,� thai p, and p, arefi".d =Ior. (rt.
lated by our choice of reference frame: p, '"
-PI)' but at tbis stage p, and p, art integra·
lion variables. It i. only apr the P. integra·
tion tbat they art restricted (P. '" -p,). and
after the IpII m
i egratiOln that they art deter·
mined by the scatterittg angle {J.
t In general, 1.At1' depend. on all
four·momenta. How",""r, in tbis case
p, "" -p, and P. '" -P" so it remains
a function only of p, and PI (assuming
a�in that spin does not come into it).
From these =tors we Can construct thrcc
scalars: p,'p, - Ipll', PI'PI = Ip,I', and
P, 'PI = Ip,lIp,lcos{J. But PI i. fixed, so the
only i'Uq;r<11io� variables on which 1.AtI' <:an
depend are Ip,1 and @.
6.3 FqlllmQt1 R"/�forQ Toy ThlOry
The integral over r is the same as in Equation 6.30, with
m2 _ m. and m] _
(E] + £1)1,2 . Quoting our previous result (Equation 6.35), I conclude that
(6,47)
where IPfl is the magnitude ofeither outgoing momentum and Ip;l is the magnitude
As in the case ofdecays, the two-body final state is peculiarly simple, in the sense
of either n
i coming momentum.
that we are able to carry the calculation through to the end without knowing the
explicit junetwnal form of .-If. We will be using Equation 6.47 frequently in later
chapters.
By the way, lifetimes obviously carry the dimensions of tilm (seconds); decay
rates If = I/f), therefore, are measured n
i n
i verse seconds. Cross sections have
2
dimensions of area - cm , or, more conveniently, 'barns':
(6.48)
Differential cross sections, dO"IdQ, are given n
i barns per steradian or simply barns
(steradians, like radians, being dimensionless). The amplitude, .-It, has units that
depend on the number ofparticles involved: if there are n external lines (incoming
plus outgoing), the dimensions of04'( are those ofmomentum raised to the power
4 - 11;
Dimensions of.,f( = (mc)·-�
(6.49)
For example, in a three-body process (A _ B + q, .-It has dimensions of
momentum; in a four·body process (A _ B + C + D or A + B _ C + D), .-It s
i
dimensionless. You can check for yourselfthat the two Golden Rules then yield the
corre<:t units for r and 0".
6.3
Feynman Rules for a Toy Theory
In Section 6.2, we learned how to calculate decay rates and scattering cross sections,
in terms of the amplitude .-lt for the process in question. Now I'll show you how to
determine .-lt itself, using the 'Feynman rules' to evaluate the relevant diagrams.
We could go straight to a 'real·life' system, such as quantum electrodynamics, with
electrons and photons interacting via the primitive vertex:
1 211
212
1 6 Tlte Feynman Calculus
This is the original, the most important, and the best understood application
of Feynman's technique. Unfortunately, it involves diverting complications (the
electron has spin t, the photon is massless and carries spin 1), which have nothing
to do with the Feynman calculus as such. In Chapter 7, I'll show you how to handle
particles with spin. but for the moment! don't want to confuse the issue, so I'm
going to introduce a 'toy' theory, which does not pretend to represent the real world,
but will serve to illustrate the melhod. with a minimum of extraneous baggage [4].
Imagine a world n
i which there are just three kinds of particles - call them A,
B, and C with masses rnA, ms, and me. They all have spin 0 and each s
i its own
antiparticle (so we don't need arrows on the lines). There is one primitive vertex,
by which the three particles interact:
-
I shall assume that A is the heaviest ofthe three and in factweighs more than Band
C combined, so that it can decay into B + c. The lowest-order diagram describing
this disintegration is the primitive vertex itself; to this there are (small) third-(lrder
corrections:
and even smaller ones of higher order. Our first project will be to calculate the
lifetime of the A, to lowest order. After that. we'll look at various scattering
processes, such as A + A --+ B + B:
XC
A
XcA �
:/ �
B
B
A + B --+ A+ B:
A
and so on.
8
6.3 Ftynmon Rullsfor 0 Ta.., ThtfJry
Fig. 6.6 A generic Feynman diagram. with external lines labeled (internal lines not shown).
Our problem is to find the amplitude Aft associated with a given Feynman
diagram. The ritual is as follows 151:
1. Notation: Label the incoming and outgoing four-momenta
Pl. Pl . . . . , p� (Figure 6.6). Label the n
i ternal momenta ql.
q2
Put an arrow beside each line. to keep track ofthe
'positive' dir«tion (folWard n
i time for external lines,
arbitrary for n
i ternal lines).
2. Vertexfactors: For each vertex. write down a factor
• .
.
-�
g is called the coupling constant; it sp«ifies the strength ofthe
interaction between A. B, and C. In this toy theory. g has the
dimensions of momentum; in the 'real·world' theories, we
shall encounter later on, the coupling constant is always
dimensionless.
3. Propagators: For each internal line, write a factor
where � is the four·momentum of the line and mj is the
mass of the particle the line describes. (Note that qj "" mj c2,
because a virtual particle does not lie on its mass shell.)
4. ConselWtion ofenergy and m(lI7�ntum: For each vertex, write a
delta function ofthe fonn
where the k's are the three four-momenta coming into the
vertex (if the arrow leads outward, then k is minus the
four-momentum of that line). This factor imposes
conservation of energy and momentum at each vertex, since
the delta function is zero unless the sum ofthe incoming
momenta equals the sum of the outgoing momenta.
[ 213
21.1 6 The
Ft)'nman Cakl.<lus
P,
A
Fig. 6.7 Lo_st.." de, contribution to A ..... B + c.
5. Integration over internal momtnta: For each internal line,
write down a factor·
and integrate over all internal momenta.
the deltajUnction: The result will include a delta
function
6. Cmcd
reflecting overall conservation of energy and momentum.
Erase this factort and multiply by i. The result is 11.
6.3.1
Lifetime ofthe A
The simplest possible diagram, representing the lowest·order contribution to
A --+ B + C, has no internal lines at all (Figure 6.7). There s
i one vertex, at which
we pick up a factor of -ig (Rule 2) and a delta function
(Rule 4), which we promptly discard (Rule 6). Multiplying by i, we get
(6.50)
This is the amplitude (to lowest order); the decay rale is found by plugging 11 n
i to
Equation 6.35:
(6.51)
• Notice (aglin) that every � g<'ts a factor of (llr) and every d gt"I. a factor of 1/(2",).
t Of coune. th� Gold�n Rul� imm�iatdy puts this ractor mock in Equation. 6.1S and 6.37, ..nd
you might wond�r why w� don't jusl keep it in .Ai. Th� prob!�m is that 1..K11. not ..K. comes
into the Gold�n Rule and th� "juan of a delta function is unddin�. So you ha"., to ...,mo,"" it
here, even though you'l! ];'<' putting it mock at the nat stage.
6.3 Feynman Rules/a, a Tay Thea".
where Ipl (the magnitude of either outgoing momentum) is
(6.52)
The lifetime ofthe A. then, is
T = - = ---
Sirfim�c
1
(6.53)
g2 1pl
r
You should check for yourselfthat T comes out with the correct units.
6.3.2
A + A _ B + B Scattering
The lowest-ordercontribution to the process A + A --> B + B is shown in Figure 6.S
In this case, there are two vertices (hence two factors of ig) one internal line.
with the propagator
-
,
two delta functions:
and one integration:
I
(211f
•
d
q
Rules 1-5, then. yield
Doing the integraL the second delta function sends q --> P. - P2, and we have
Fig. 6.8 Lowest·order contribution to A + A ..... B + B.
1 21S
2161 6 Th, Ft>tnmon Co/",/us
')f a
A�
"1
Y,i,
A
c
�B
Fig. 6.9 S�ond diagram contributing in lo_st order to A + A ...... B + B.
As promised, there is one remaining delta function. reflecting overall conservation
ofenergy and momentum. Erasing it and multiplying by i (Rule 6). we are left with
(6.54)
But that's not the whole story. for there is another diagram of order g2. obtained
by 'twisting' the B lines (Figure 6.9).* Since this differs from Figure 6.8 only by
the interchange Pl P4, there is no need to compute it from scratch; quoting
Equation 6.54, we can write down immediately the total amplitude (to order g2) for
the process A + A ---Jo B + B;
....
(6.55)
Notice, n
i cidentally, that A s
i a Lorentz-invariant (scalar) quantity. This is always
the case; it is built into the Feynman rules.
Suppose we are n
i terested in the differential cross section (dO" jdQ) fOr this
process, in the eM system (Figure 6.10). Say, for simplicity, that /tIA /tIg /tI and
mc-O. Then
- pd m�,2 P! pi
(Pl - PI)2 m�c2 � p�
(p.
- -
-
=
+
- 21'2 . P4 = _2p2(1 - cosO)
(6.56)
-
=
+
- 2Pl . 1'2 = _2p2(1 +cosO)
(6.57)
(where p is the incident momentum ofparticle I), and hence
g'
A= pI sin2 0
--
According to Equation 6.47, then,
'
do 1
"'<'
=
dQ i 16nEp2sin20
(
)
(6.58)
(6.59)
(there are two identical particles in the final state, so 5 = Ij2). As in the case of
Rutherford scattering (Example 6.4), the total cross section is infinite.
• You don'l get � another n..w di.agram by twisting the A lines; the only choice here i$ whether
Pl 'onn�ts 10 PI or to 1'1-.
6.3 Fqnmon Rulesfor ° Toy Theory
/
.�7{(
_ _ .
��
Before
Fig.
6.10 A + A _ 8 + 8
B
in the
eM fr�me.
B
After
6.3.3
Higher-order Diagrams
So far we have looked only at lowest-·Qrder ('tree level·) Feynman diagrams; in the
case ofA + A ...". B + B, for instance, we considered the graph:
CX
B
B
A
This diagram has two vertices, so .,(( is proportional to g1. But there are eight
diagrams withfour vertices (and eight more with the elCternal B lines 'twisted'):
• five 'self.energy' d
iagrams, in which one of the lines sprouts
a loop:
A
cc
B
A
A
B
B
• two 'vertex corrections', n
i which a vertex becomes a triangle:
• and one 'box' diagram:
c
M
C
B
B
A
A
B
1 217
2111 6 Tht FeynmQn C"iwius
don't count.)
I am certainly not going to evaluate all these 'one·loop· diagrams (or even think
about two-loop diagrams), but I would like to take a closer look at Ont ofthem - the
one with a bubble on the virtual Cline:
A-?;:"
p,
Applying Feynman roles 1-5, we obtain
Integration over ql using the first deltafunction, replaces ql by (PI - PI); integration
over q., using the last delta function, replaces q. by (p. - 1'2):
'
g'
[(PI PI)l m2cl)[(P. P2JZ m2cZ]
�·(Pl -PI - q2 - ql) �·(q2 + ql - p. + 1'2 )
d4q2 d4ql
x
(� m�cl)(q� m�c2 )
J
(6.61)
Here. the firstdelta function sends q2 -+ PI - PI - qJ, and the seconddelta function
becomes
�4 (P1 + Pz - PI - P4)
which, by Rule 6, we erase, leaving
1
p,
q)2 m�c2](ql _ mic2)
d'q
(6.62)
(I drop the subscript on qJ at this point.)
6.3 Fryrrm"rr Rulnforo TO)' Thoro"..
You can try calculating this integral, if you've got the energy, but I'll tell you
right now you're going to hit a snag. The four-dimensional volume element
= dq dQ' (where dQ' stands for the angular part), just
could be written as
as in two-dimensional polar coordinates the element of area is r dr de and in
three-dimensional spherical coordinates the volume element is ,; dr sin 8 d8 d¢'.
At large q the integrand is essentially just Ijq·, so the integral has the form
d"q ql
fOO q-oql l dq
= In qlOO = 00
q
(6.63)
The n
i tegral is logarithmically divergent at large q. This disaster, in one form
or another, held up the development of quantum ele<:trodynamics for nearly two
decades, until, wough the combined efforts of many great physicists - from
Dirac, Pauli, Kramers, Weisskopf, and Bethe through Tomonaga, Schwinger, and
Feynman - systematic methods were developed for 'sweeping the infinities under
the rug'. The first step is to regularize the integral, using a suitable cutoffprocedure
that renders it finite without spOiling other desirable features (such as Lorentz
invariance). In the case of Equation 6.62, this can be accomplished by introducing
a factor
(6.64)
under the integral sign. The cul<Jff mass M is assumed to be very large, and will be
taken to infinity at the end ofthe calculation (note that the 'fudge factor', Equation
6.64, goes to 1 as M -+ (0).* The integral can now be calculated [6J and separated
into two parts: a finite term, independent of M, and a term involving (in this case)
the logarithm of M, which blows up as M -+ 00.
At this point, a miraculous thing happens: all the divergent, M-dependent terms
appear in the final answer in the form of additwns I<J the masses and the coupling
constant. If we lake this seriously, it means that the physical masses and couplings
• No one would deny that this procedure is artificilll. Still, it can k argu� that the inclusion of
Equation 6.6-4 merely confesses our ignorance of the high..mergy (short distance) khavior of
quantum field theory. Perhaps the Feynmm propagators are oot quite right n
i this r.-gime, and
M is simply a cnlde way of accounting for the unknown modification (Ibis would bf, the cal>'",
for """-mple, if the '�rtides' have substructure that becomes relevant at extremely dose range.)
Di...c said, of renormalization,
It's just a stop-gap proudure. There must be some fundamental
change in our ideas, probably a changejust asfundamental as flu
passage from Bohr's orbit thtory I<J quantum mechanics. When
you get a number turning out to be infini� which ought I<J be
finik, you should admit that there is somelhing wrong with your
equatioltS, and not hope thaI you can gel a good theory just by
doctoring up that number.
P. Buddey and F. D. Peat. A �tion ofP"�iaI (Toronto: University of Toronto Press, 1979),
page 39.
1 219
220
1 6 Th' Ff)'nman Col,ulus
are not the m's and g's that appeared in the original Feynman rules, but rather the
'renormalized' ones, containing these extra factors:
(6,65)
The fact that lim and .5g are infinite (in the limit M ___ (0) is disturbing, but not
catastrophic, for we never measure them anyway; all we ever see n
i the laboratory
are the physical values, and these are (obviously) finite (evidently the unmeasurable
'bare' masses and couplings, m and g, contain compensating infinities): As a
practical matter, we take account of the n
i finities by using the physical values of
m and g in the Feynman rules, and then systematically ignoring the divergent
contributions from higher·order diagrams.
Meanwhi
l e, there remain thefinik (M,independent) contributions from the loop
diagrams. They, too, lead to modifications in m and g (perfectly calculable ones, in
this case) - which, however, are functions of the four·momentum of the line in
which the loop is inserted (Pl - Pl in the example). This means that the effictive
masses and coupling constants actually depend on the elUrgles of the particles
involved; we call them 'running' masses and 'running' coupling constants. The
dependence is typically rather slight, at low energies, and can ordinarily be ignored,
but it does have observable consequences, in the form of the Lamb shift (in QED)
and asymptotic freedom (in QCD).t
• In case it is some comfort. essentiaUy the same thing occurs in dassicnl eJe.:trodynamics: the
dectrostatic �n�rgy of a point charS'" is n
i finl!�, and makes an infinil� contribution (via E ='
"...2) to 1M �rtide's mass. Per�ps this means that there are no true point charges, n
i classi.
cal electrodyn.a.mics; perhaps thafs what it means in quantum field theory. too. In neither case.
�, do _ know how to avoid th� point particle :os a theoretical construct.
i QED and QCD w:os suggested in
t A physical interpretation of the running ooupling constant n
Chapter 2, Section 2.3. A nice explanation of mass renormalization is given by P. Nelson in
Amerialll Scim!isI, 73, 66 (l98S):
According to renormalization Iheory, not only the strengths
ofthe variow inkrae/ions but the massts of the participating
particles appear to vary on differing length scales. To get a
Jed for this sumingly parado:dcal swl<:ment, inwgine firing a
cannon underwater. Even neglecting.friction, the trajectory will
be very diffortnt from the ccrresponding one on land, since the
cannonball must now drag with it a considerable amount of
wakr, modiJYing its apparent, or "eJfective," mass. We can ex·
perimenwlly measure the cannonball's eJfectiVil mass by shaking
it to andfro at a fak w, computing the massfrom F = mao
(This is how astronauts ''weigh'' themselves in space.) Hav·
ingfound the effective mass, we can now rtplace the difficult
problem of underwakr ballistics by a simplified approximation:
wt: ignore the wakr altogether, but in Newton's equations wt:
simply replace the trut cannonball mass by the effective mass.
The complicaud details of the inkraction with the medium are
6.J Feynmon Rules for 0 Toy Theory
The procedure I have sketched in the last three paragraphs is called renor·
ma/izatiolt [71. If all the infinities arising from higher·order diagrams can be
accommodated in this way, we say that the theory is renormalizable. ABC the­
ory and quantum electrodynamics are renormalizable. In the early 19705, 't Hooft
showed that all gauge theories, including chromodynamics and the electroweak the­
ory of Glashow, Weinberg, and Salam, are renormalizable. This was a profoundly
important discovery, because, beyond lowest-order calculations, a nonrenormaliz·
able theory yields answers that are cutoff-dependent and. therefore, really, quite
meaningless.
References
1 S��, for exampi�. Tayior. J. R.(2005)
Oassical M,,"""ics. University Sci·
ence Books. Sausalito, c.A. Sect.14.6,
or; (a) Goldstein, H., Poole, C.
and Safko, J. (2002) CWssicaJ Mo·
,"""ies. 3rd edn. Addison·W61ey,
1 Se�, for exampl�. Puk, D. (1992)
Inlr�udion Ie Ih, QlUllllum The­
San Francisco, C.A. Sect. 3-10.
ory, 3rd edn, McGuw-Hill, New
York. Sect. 7.9; la) Townsend, J. S.
(2()(10) A MO<km Approach 10 Quail.
lum Mec"""ics. University Scienc� Books, Saus.aHto, C.A Sect.
IoU; (b) Griffiths, D. /. (200S) III'
J. M. and Rohrlich. F. (1976) Th,·
01)' of Pholons and Eleclrons. 2nd edn,
Springer-Verlag, Berlin. Sect. 8_6; (b)
Hagedorn. R. (1963) Relalivistic Kine­
malics. Benjamin, New York. Chap.
7; (e) Ryder. L. H. (1985) Quantum
Field The0l)'. Cambridge University Press, Cambridge. Sect. 6.10;
(d) P6kin. M. E. and Schroeder.
D. V. (1995) An Introduction to
Qi<atllum Field Thtory, Perseus,
Cambridge. Chap. 4; (�) Weinb�rg.
S. {199S) The QlUllllum Theory of
Fields, 1'01. I, Cambridg� University Pr�ss. Cambridge. Sect. 3_4.
Irodudioll 10 QlUlntum Mcch<>n-
4 This model w�s shown to me by
ies, 2nd edn. Prentic� Hall, Up-
tic�ted tre�tment see Aitchison.
per Saddl� River, N.J. Se<:t. 9.2.3.
3 Convincing and accessible accounts
are in fact difficult to find. Feynman,
R. P. (1%1) Theel)' of Fulldamen-
10.1 Processes, Benjamin, Readn
i g,
M.A. Chapt�n IS and 16, is a good
place to start. See also; (a) Jauch,
Max Dresden. For a mOre sophis­
Caugr: l1u:OM ill Partiek: Ph}'l'ics,
!. J. R. �nd Hey. A. J. G. (2003)
3rd edn, 1'01. I. Institute of Physics.
Bristol, UK. Section 6.3.
S You may well ask wher� th�se rules
come from, and I am not Sure Feyn­
man himself could have given you
thus reduced to cktumining one eff(£tive paramell:r.
A key feature of this approach is Ihat /he effictive mass so cem­
puttd ckpends on tv, sinu as w approaches zuo, for example,
the water has no effect whatevtr. In other words, the presence o
fa medium can introduce a scak-depencknt eff(£tive
mass. We say that lhe ejftctive mass is "renormalized" by the
medium. In quantum physics, every partick moves through
a "medium" consisling of tk quantumfluctuations of all
particles prestnt in the theory. We again tau into aCCOWlt
/his medium by ignoring il but changing the valllM of our
parameters UJ sca!e.ckpendent "ejftclive" values.
1 221
222 1
6 The F�"mfm Clllcu!us
� compl�t�ly satisf�ctory answ�r in
1949, wh�n h� fiTSt published the
rules for QED Feynm�n, R. P. (19�9)
Physical Review. 76, 7�9·769. It reo
mained for Freeman Dyson to show
how the Feynm.an rul�s could be ob·
tained from quantum field theory;
(a) Dyson. F. ). (1949) Physic,,1 Re·
vitw 75, -486 �nd 1376; �nd (19S1) 82.
428. For a fascinating personal his·
tory of these events, see Chapt�n S
and 6 of Dyson's book (1979) Dis·
tu,;,;� T1u: U";"",,e, Harper & Row,
New York. W� shaJ.l return to the
question of how Feynman·s rules
a� derived in Chapt�r 11; for now I
will simply tr�at them as axioms.
, The method is expbin�d in Sakurai.
J. ). {1967} Advonud QUIln!um Me·
cha"iCl. Addison·Wesley. R�ading.
M.A.; see particularly the useful col·
lection of formulas in Appendix E.
? Renomuliz.ation of ABC the-
ory is studied in Griffiths. D. ).
and Kraus. P. (1992) America"
Journal of Physicol, 60. 1013.
Problems
U Derive Equation 6.3.
(Hin!: What fraction ofth� origin�l sample deca)'!i between t and
t + dt? What, then, is the (initial) probability, PIt) dt, of any given particle deaying
be�n t and t + dt? The avoage lif�time is fo"" tp{t) dt.)
'.2 Nuclear ph)'!iicists tradition�lIy wor!< with 'half·life' (t,o) instead ofmean life (r); I,n is
the time it takes for halrth� members ofa large sample to decay. For exponential decay
(Equation 6.2). derive the fonnula for " ll {as a multiple of rl·
6.3 (a) Suppose you staned out with a million muons (at rest): how many would still be
around 2.2 x IO-s seconds later?
(b)Wmot is the probability ofa ",- lasting mOr� th.m 1 second (express your answ�r as a
poweroflO)?
U A nonrelativistic panicle ofmass m and (kinetic) energy E scatters from a fixed repulsive
potential, VIr) = klrl, where k is a constant.
(a) Find the scattering angle. 8. as a function ofthe impact panmeter. b.
(b)De\ermine the differential cross section dO" {dO, as a function of8 (not b).
(e) Find the total crou section.
(References: Goldstein. H.. Poole. C. �nd Safko, ) . (2002) CkrniroJ Mecha"ic" 3rd Mn,
Addison·Wesley. San Francisco, CA. Sect. 3·10., Equation 3·97; Becker, R.A. (19)�)
1,,'rodWCI;OII to Thwrdirol Mu;ha"iCl, McGraw·Hill, New York, Example 10-3.)
6.5 Derive Equation 6.34. starting (rom Equation 6.31 with u= mJc.
6.6 As an application of the Golden Rule. consider the decay of]f° ...... Y + y. Of course.
the ]f° is a composite object, so Equation 6.35 does not really apply, but l�t's pretend
tlut ifs a true �lem�ntary particle, and see how dos� _ com�. Unfortunately, we don't
know the ampiitude..R'; however, it must move the dimensions of mass times velocity
(Equation 6.�91. and there is only one mass and one ve!ocil)' available. Moreover. the
emission of each photon introduces a factor of";;; (the fine structure constant) into
A. as we shall sec in Chapter 7, so the amplitude must be proportional to "'. On this
basis. estimate the lifetime ofthe 11°. Compare the experimental value. (Evidently. the
decay of the ",0 is a much more complicated process tnan this crude model suggests.
See Quigg, C. (1997) Caugr Thwries oJthe Strong. Weak, a.w Elutrom'W'Ctic Inkract;ons.
Addison·Wesley. Reading, M.A. Equation 1.2.2S - but beware ofthe misprintf should
�
be squared.]
6.7 (a) �rive Equation 6.�1 for scattering ofparticles I and 2 in the CM.
(b)Obtain the corresponding formula for the lab frame (particl� 2 at rest).
6.3 Fey"mcl" Rulesfor " Toy Theory
6.8 Consider �l�sticsc.atkring, a + b -+ ,, + b, in the l�b fr�me (b initially �t rest), assuming
the target is so h�avy (m�cl » Eo) th�t its recoil s
i n�gligibl�. Determin� the differ�ntial
SGlnering cross section. [Hint: In ths
i limit the lab fr�me and the CM fr�m� �n: the
same.)
6.� Consid�r the collision 1 + 2 ..... 3 + 4 in the lab frame (2 at r�st), with particles 3 and 4
m<lSsl�ss. Obt�in the formula for th� diff�r�ntial cross section.
6.10 (.)Analyz� the problem of elastic SGln�ring (m3 = m"m, = ml ) in th� bb fr.lm�
(p;lrtid� 2 at rest). Derive the formub for th� diff�r�ntial cross section.
(h)lfth� incident p;lrtide is massless (ml = 0), show that th� result in p;lrt (a) simplifies
"
6.11 (0) Is A ..... B + B � possibl� process in the ABC theory;>
(b)Suppos� � diagram has "'" external A lines, II! ext�rnal B lines, and lie external C
lines. Devdop � simple criterion for determining whether it is an allowed re�ction.
(e) Assuming A is heavy enough. what are the next most likely dec�y modes, �fter
A -+ B + Q Draw � Feynm�n di:lgr�m for each d<'C�y.
6.12 (.) Draw �1I the lowest·order diagrams for A + A ..... A + A. (There are six of them.)
(h) Find the amplitude for this process. in lowest order. assuming m, ., me _ o. leave
yOUl answer in the form of an integral over one rem�ining four·momentum, 'I.
6.U Calculate daIdQ for A + A ..... B + B. in the CM fr�me, assuming ma
me O. Find
the total cross section. q.
6.1' Find do-/dQ and (1 for A + A -+ B + B in the lab frame. (let E be the energy. and p the
momentum. ofthe incident A. Assum� mB me 0.) Determine the nonrelativistic
�nd ultrarelativistic limits ofyOU! formula.
6.1� 1'1 Determine the lowest·order �mplitude forA + B -+ A + B. (Then: are IW<>diagr�ms.)
[b)Find the differential cross section for this process in the CM fr;o.me. assuming
m... m, m, me o. Express your answer in terms ofthe n
i cident energy (of AI.
E, �nd the scattering �ngIe (for p�rtide A). 8.
Ie) Find daIdQ for this process in the lab frame. assuming B is much heavier than A and
remains stationary. A is incident with energy E. [Hi,,!: See Problem (6.8). Assume
mB » m..., me, �nd fle1.]
(d)ln case (e). find the total cross section. q.
=
=
=
=
=
=
=
1 223
7
Quantum Electrodynamics
[11 ths
i chap'u I introduce the Dirac equation, statt the Ftynman rules fer quantum
tkctrodynamics. develop ust:fu! calculational tools, and dtrivt SOIm of the classic QED
results. 11u: treatment !tans hwvily on makrialfrom Chapters 2, 3, and 6, as weU as the
5pln-1 formalism in Chapter 4, In turn, Chapter 7 is the irniispmsab!e foundation for
everything thatfollows (however, you may want to skip Example 7.8 and SUlion 7.9,
togellur with the rdattd passages in Chapttrs 8 and 9).
1.1
The Dira.;: Equation
Although the 'ABC' model in Chapter 6 is a perfectly legitimate quantum field
theory. it does not describe the real world, because the particles A. B. and
C have spin 0, whereas quarks and leptons carry spin
and mediators carry
spin L The inclusion of spin can be algebraically cumbersome; thaI's why [
introduced the Feynman calculus in the context of a 'toy' theory free of such
distractions.
In TWnrelafivinic quantum mechanics, particles are described by Schrodinger's
equation; n
i relativistic quantum mechanics, particles of spin 0 are described by
the Klein- Gordon equation, particles of spin � by the Dirru equation, and particles
of spin 1 by the Proca equation. Once the Feynman rules have been established,
�
however, the underlying field equation fades into the background - that's how we
got through Chapter 6 without ever mentioning the Klein-Gordon equation. But
for spin the very notation of the Feynman rules presupposes some familiarity
with the Dirac equation So for the next three sections we'll study the Dirac theory
!
.
in its own right.
One way to 'derive' the 5chrodinger equation is to start with the classical
energy-momentum relation:
(7.1)
....
In<rod�,lion I<> E!e �"''Y P�ni.ck>. S«onJ Editi",," o..vid Griflitlu
Copyrisht 0 2008 WILEY·YCH V'TbS GmbH & Co. KGaA. W�inMim
ISBN: 973-}·S27·.0601·2
226
I 7 Quan!IJm Electrodynamics
apply the quantum prescription
p ...... -ifiV,
. ,
E ...... fi
l-
(7.2)
"
and let the resulting operators act on the 'wave function', W:
(Schrooinger equation)
(7.3)
The Klein-Gordon equation can be obtained in exactly the same way, beginning
with the relativistic energy-momentum relation. E2 - p2C1 = m2 C, or (better)
(I'll leave out the potential energy. from now on; we'll stick to fru particles).
Surprisingly, the quantum substitution (Equation 7.2) requires no relativistic
modification: in four-vector notation, it reads
plJ. ...... ifi illJ.
(7.5)
,IJ. �
­
-
(7.6)
Here'
,
ilx!'
which is to say
ill =
,
(7.7)
ax'
Putting Equation 7.5 into Equation 7.4, and letting the derivatives act on a wave
function ",,t we obtain
(7.S)
(Klein-Gordon equation)
(7.9)
Schrodinger actual!y discovered this equation even before the nonrelativistic one
that bears his name; he abandoned it when (with the Coulomb potential included)
• The grldienl with respect to a wnt",,,,,rianl position·time four·vrdor "" is itself a covariant
four·v�tor; hen<:e the placement of the index. Writlen out in full, Equation (7.5) $.1)'$
(Elc. -p) ..... ih(��. V). Of course. ,,�_ a/h�. � Problem 7.1.
t In nonrelativistic quantum m�anics we customarily use the capital ktll':r ("-') for the wave
function, and rese� t for its spo.tial �rt (Equation 5.3). In the relativistic theory it is more
common to use 'II for the wa� function itself.
7. T TM Dirac Equation
it failed to reproduce the Bohr energy levels for hydrogen. The problem is that the
electron has spin and the Klein-Gordon equation applies to particles with spin O.
Moreover, the Klein-Gordon equation is incompatible with Born's statistical inter·
pretation, which says that 11fr(r)12 gives the probability of finding the particle at the
point r. The source of this difficulty was traced to the fact that the Klein-Gordon
equation is second order n
i t.' So Dirac set out to find an equation consistent with
the relativistic energy-momentum formula, and yet first order in time. Ironically,
in 1934, Pauli and Weisskopf showed that the statistical interpretation itself must
be refonnulated n
i relativistic quantum theory,t and restored the Klein-Gordon
equation to its rightful place. while keeping the Dirac equation for particles ofspin
l,
�.
Dirac's strategy was to ·factor' the energy-momentum relation (Equation 7.4).
This would be easy ifwe had only pO (that is, if p were zero):
(7.10)
We obtain two first·order equations:
(p° - mc) = O or
(p° + me) = O
(7.11)
either one of which guarantees that 1'"PI' - ml,2 = O. But it's a different matter
when the spatial components afe included; in that case we are looking for something
ofthe form
(7.12)
where fj� and yA are eight coefficients yet to be determined.f Multiplying out the
right-hand side, we have
We don't want any terms linear n
i p�, so we must choose W = y"; to finish the
job, we need to find coefficients y" such that
• Notice that the SchrOdinger equation is first order in t.
t A rdotivisti( thMry has to account for pair production and annihil<otion. md hence the number
of particles is not a conse",...! qU.1ntity.
t In case the noution confuses you, let me write Equation (7.U) 'Iong·hand·:
(p0)2
_
(p1)2
_
<i )2
_
(pl )l mlcl = (fjOpO plpl
_
_
x
_
(yOl - yip!
fjlpl
_
_
y l}f
fjlpl + me)
_
y lpl - me)
1 227
2281
7 Ql/QnUlm Electrodynamics
which is to say
(l)2 (i)2 (p2)2 (p1)2 = (yO)2(p0)2 + (y l )2(pl )2 + (y2)2(p2)1
O ! !
+ (yl)2(pl)1 + (y y + y yO )PoPI
_
_
_
+ (yOyl + y2 yO)PoPl + (yO yl + yl yO)PoPJ
+ (y l y 2 + y 2yl )P1Pl + (y l y l + y lyi)P1Pl
(7.13)
You seethe problem: we couldpick yO = 1 and y l = y2 = yl = i, but there doesn't
seem to be any way to get rid ofthe cross terms.
At this point Dirac had a briUiant inspiration: what ifthe y 's are matrices, instead
of numbers? Since matrices don't commute, we might be able to find a set such that
y"y' + y.y" = 0,
for iJ. oF u
(7.14)
Or, more succinctly,
ly",y 'J = 2g'"
(7.15)
where g"" is the Minkowski metric (Equation 3.13), and curly brackets denote the
anlicommutator:
IA. Bl =: AB + BA
(7.16)
You might try fiddling with this problem for yourself. It turns out that it can be
done, although the smallest matrices that work are 4 x 4. There are a number of
essentiaUy equivalent sets of 'gamma matrices'; we'll use the standard 'Bjorken
and Drell' convention [1):
(7.17)
where ui(i = 1,2, 3) is the indicated Pauli matrix (Equation 4.26). 1 denotes the
2 x 2 unit matrix, and 0 is the 2 x 2 matrix of zeroes.'
• When the context allows no Toom for ambiguity. 1"11 uSC 1 and 0 this way for 2 x 2 QT 4 x 4 rna·
trices; also, a unit m.1lrix of the appropriate di""'nsion is ;mplkd, when necessary. as on the
right.hand side of Equation 7.1S Incidentally. since (1 is nO( the sp;itial part of a {QUT·vector. we
do not distinguish up� and lower indices: ,, ' . "i.
7.2 So/uliol1s 10 Iht Dirac fquoliol1
a 4 x 4 matrix equation, then, the relativistic energy-momentum relation
factor:
As
d«s
(7.18)
We obtain the Dirac equation, now, by peeling offone term (it doesn't really matter
which one, but this is the conventional choice - see Problem 7.10):
yl'pl' - ltIl: = 0
(7.19)
Finally, we make the quantum substitution PI' _ iii. 01' (Equation 7,5), and let the
result act on the wave function 1/r:
(Dirac equation)
(7.20)
Note that 1/r is now a four..element column matrix:
(7.21)
We call it a 'bi-spinor', or 'Dirac spinor'. (Although it has four components, this ob­
ject is 110t a four-vector. In Section 7.3 I'll show you how it does transform when you
change n
i ertial systems; ii's not going to be an ordinary Lorentz transformation.)
7.2
Solutions to the Dirac Equation
Suppose that 1/r is independent ofposition:
o1/r = a1/r = a1/r = 0
ax ay oz
(7.22)
In view of Equation 7.5, this describes a state with zero momentum (p = 0), which
is to say, a particle at rest. The Dirac Equation (7.20) reduces to
iii. oo1/r
a,
,
- y - - mc
where
(�
1/r = O
(7.23)
(7.24)
(7.25)
1 229
230
I
7 QllQnlum EitClfOdynamics
consists of the upper two components, and
(7.26)
comprises the lower two. Thus
(7.27)
and the solutions are
(7.28)
Referring to Equation 5.10, we recognize the factor
(7.29)
as the characteristic time dependence of a quantum state with energy E. For a
particle at rest, E = mc1, so 1/1A is exactly what we should have expected, in the
case p = O. But what about 1/Is? It ostensibly represents a state with negativt: energy
(E:= -mel). This is the famous disaster I mentioned back in Chapter 1, which
Dirac at first tried to avoid by postulating an unseen infinite 'sea' ofnegative.energy
particles, which fill upall those unwanted states: Instead, we now take the solutions
with 'anomalous' time dependence to represent antiparticles with p�itivt energy.t
Thus 1/1A describes electrons (for example), whereas 1/18 describes positrons; each s
i
a two·component spinor, just right for a system of spin j. Conclusion: The Dirac
equation with p = 0 admits four n
i dependent solutions (ignoring normalization
factors, for the moment):
(7.30)
• You might ask why we don't simply stipulate that t.{O)
'" 0 - call thr 'negati�""rgy' solu·
lions 'physically unacceptable', and forget about them. Unfortunately, this can't be done. In a
qu..ntum system we need a compiete set of states, and the positivr rnrrgy statrs by themselves
arc not complete.
t In the Schr(ldinll"'" equation the sign of i is purely conventional. Had S<:hrOdinger made the op­
posite choice. tEl'" would be the 'normal' time drpendrncr for a slalionary state ofenergy E. In
the relativistic thCQry bark signs arise, and this. when properly interpreted, implies the existence
of antiparticlrs.
7.2 �lufiortS10 th, Diroc Equotion 1 211
They describe, respectively, an eltttron with spin up, an eltttron with spin down,
a positron with spin down: and a positron with spin up.
We look next for plant-WQ\Ie solutions;t
(7.31)
We're hoping to find a four-vector kl' and an associated bispinor u(k) such that
1,/f(x) satisfies the Dirac equation (a is a normalization factor, irrelevant to our
present purpose but necessary later to keep the units consistent), Because the x
dependence is confined to the exponentt
(7,32)
Putting this into the Dirac Equation (7.20), we get
(/iyl'k" - mc) u 0
(7.33)
=
Notice that this equation is purely algebraic - no derivatives. If u satisfies
Equation 7.33, then oJ! (Equation 7.31) satisfies the Dirac equation.
Now
0
0 ) _k. (
-1
-0"
u
) (
0
_
-
If
k · O"
-k U
-If
)
(7.34)
) ( )
(_tik° - me)
UB
-iii<
U
"'
• Notice the 'backward' antiparticle spin ori�ntation" In Dirac'. in!erpr�tl.tion (which rem.1im a
hmdy mnemonic de>ice) >I'(ll is a �li"'·energy tlutron sme with spin up, whose "OStota (a
'hol�' in th� 'sea') brhavr. as a I""ti
i ...�nergy po>itron with spin down (21.
t H�r� k x k�x" �<"I- k· r, so the !Nlpart of the exponenti.J.I is COS(�CI - k · r). which
represents a sinusoidal pl.;one wavr of (angular) frequency w = ck" and wa�lffigth � .. 2"'/lkl,
propagating in 1m, direction k.
:i This looks right. but if it makes you nervous you (an easily check it:
=
=
iloe-ib
=
(l/c)�e-j�<I+ik-r
(and � = 1:0). Similarly
(but k' '"' 0,1
-
.
"
=
_ikOe-i.l:-x
232 1 7 QuaMlum ElearadyMamiC3
where, as before, the subscript denotes the upper two components, and B stands
for the lower two, [n order to satisfY Equation 7.33, then, we must have
A
(7.35)
Substituting the second ofthese into the first gives
(7.36)
But
Ik. -
' 'I)
-k,
(7.37)
)
.l:r(.l:x - iky) - .l:z(.l:x - jky) = k2l
(k,. + i.l:1)(kx iky) + ki
-
(7.38)
where 1 is the 2 x 2 unit matrix (written in explicitly, just this once). 1bus
k'
(7.39)
and hence'
(7.40)
In order for 1/1 = exp(-ik . x)u(.I:) to satisfY the Dirac equation, then, fikI' must be
a four-vector, associated with the particle, whose 'square' is m2,2. Of course, we
know such a quantity: the energy-momentum four-vector. Evidently
k" = ±p"In
(7.41)
The positive sign (time dependence t-iE" ,,) is associated with particle states, and
the negative sign (time dependence t+iE" �) with antiparticle states.
Returning to Equaiton 7.35 (and using Equation 7.37). it is a simple matter to
construct four independent solutions to the Dirac equation:
.
(1) PIC
k UA =
(t)
0
: UB =
:0-p o
()
1
y + m' 0
�
'
,
.
-
( )
p,
.
E + m�- P",+ !PY
m� argument, starting
• Equation 7.19 would also allow u... ., 0 as a solution. H��r. th� ..
with Equation 7.35 but inserting the first into the se.::ond. yields Equation 7.39 with "I in place
of ".... So. unless u... and '" are both zero (in which C
a
... we IoIve no solution at all) Equation
7.40 must hold.
(2) Pick ",
.
(3) PICk Us
•
:=
()
()
0 , "• . �
1
1
o
+ mc
y
: U,o. =
p.•
.
-
p + mc
(0)
()
1
1
7.2 Solulion< 10 Ihe Diroc Equot;o"
'
,
. _
E + mc
'
( )
( )
P. - iP'
-pz
p.
E + mc, P% + lpy
0 • --.
For (1) and {2} we were obliged to use the plus sign in Equation 7.41 - otherwise
Us blows up as p --l> 0; these are particle solutions. For (3) and (4) we must use the
minus sign; these are antiparticle states.
A convenient nonnalization for these spinors is·
{7A3}
Here the dagger signifies the transpose conjugate (or Hennitian conjugate):
'0
(7.44)
With the resulting normalization factor (Problem 7.3)
(7.45)
" ' . N[ ,J.! )
the four canonical solutions become:
E + mc1
c(p� + iFy)
E + mc1
.
• Noticr that any multiple of u i. still a solu·
tion to Equation 7.33; norm.1Hzation merrly
fixes the overall constant. Actually. there are
at least three different con�ntions in the
literature: "t,,:o 2EI' (Halzen and Martin).
(7.46)
"tu '" Elm<l (Biorken and Drell). "T" .. 1
(Bogoliul.>ov and Shirkov)_ In this one in­
stance I dq>art from Bjorken and DreU.
whose choice introduces spurious diffiOllties
whrn m -+ O.
1 233
2),(
.. ['�' - ) [ '�.I )
I 7 Q�Q"I m £I�drody"Qmics
''''I
E + me2
II)
v =N
e(-P%)
E + mel
o
'
vll) = -N
E + mel
e(p", + ipy)
E + me2
(7.47)
1
o
1
(7.48)
It s
i customary, from here on, to use the letter v for antiparticles (and to include
a minus sign in vm), as n
i dicate<!. Notice that whereas particle states satisfY the
momentum space Dirac equation (see Equation 7.33) in the form
(y"p" - me)u = 0
(7.49)
antiparticles (v's) satisfy:
(y"p" + me)v = 0
(7.50)
You might guess that ull) describes an electron with spin up, I'll) an electron
with spin down, Vii) a positron with spin up, and v 12) a positron with spin down.'
but this s
i not quite the case. For Dirac particles the spin matrices (generalizing
Equation 4.21) are
,
(7.51)
i
and it's easy to check that I'll), for instance, is not an eigenstate of 1:%. However. f
we orient the 2 axis so that it points along the direction of motion (in which case
l)
p� = py = 0) then 1'11) , 1'121 , v ii ) and v l ) �re eigenspinors of Sz; I'l and Vii) are spin
up, uPI and tlill are spin down! (Problem 7.6).
l
Incidentally, plane waves are, of course, rather special solutions to the Dirac
equation. They are the ones of interest to us, however, because they describe
particles with specified energies and momenta, and in a typical experiment these
are the parameters we control and measure.
• Stt th� fOOb1.01� 10 EqlL1.tion 7.30 for positron
spin orientations.
t As � man�r of facl, il is j"'pouibk 10 con·
StruCt plane·wave solutions to th� Dirac
equation and ue. at the pme time. eigen·
states of S, (except in the special case
p = p,t). The reason is that S by itself is not
a conserved quantity: only the 101<>1 angular
momentum. L + S. is conserved here (s�
...
Problem 7.81. It is possible to construct
ei�nstat of kdicily, 1: . 1> (there's no orbil<ll
angular momentum about th� direction of
motion). but theu- are rather cumbersom�
(� Problem 7.7), and in practice il is easier
to work with the spinars in Equations 7.46
and 7.47. even though their physical interpre·
tation is not so clean. AU that rwlIy matters s
i
thai � !rave a c.ompl& set of solutions.
7.3 BiIi,.,�orCOllOrjo"ts
7.3
Bilinear (ovariants
! mentioned in Section 7.1 that the components of a Dirac spinor do not transform
as a four·vector, when you go from one n
i ertial system to another. How, then, do
they transform? ! shall not work it out here (you get to do it, in Problem 7.11), but
merely quote the result: ifyou goto a system moving with speed v in the x direction
(7.52)
1/1 - 1/1' = 51/1
where 5 is the following 4 )( 4 matrix:
5 = a+
+
a_ yOy l =
( a+
"_O'l
,_.,
'+
with
a
±
)
�
r
0
0
,-
0
0
'+
,-
0
0
,-
'+
)
'
:
'+
= ±J!(Y ± I)
(7.S3)
(7.54)
and y = l/Jl v1/c2, as usual.
Suppose we want to construct a scalar quantity out of a spinor 1/1 . It would be
reasonable to try the expression
Unfortunately, this s
i not invariant, as you can checkby applying the transformation
rule:"
(7.56)
, No� wt the tran.post of. product is the product of the transposes ;n "'","'" orlkr:
The ume goes for tile Hermitian conjugate,
1 235
2361 7 Quam".., EI,ctrodynamics
In fact (Problem 7.13):
StS = S2 = y
(
1
(7.57)
-(v/c)UJ
Of course, the sum ofthe squares of the elements of a four·vector is not invariant
either; we need minus signs for the spatial components (Equation 3.12). With
a little trial and error you will discover that in the case of spinors we need
minus signs for the third and fourth components. Just as we n
i troduce<! covariant
four-vectors to keep track of the signs in Chapter 3, we now introduce the adJoint
spinor:
(7.58)
I claim that the quantity
s
i a relativistic invariant. For st yOS
= yO (Problem 7.13). and hence
(7.59)
(7.60)
In Chapter 4 we learned to distinguish scalars and pseudoscalars, according
to their behavior under the parity transformation, P; (x, y, z) -+ (-x, -y, -z).
Pseudoscalars change sign; scalars do not. It is natural to ask whether ifi1/1 is the
former type. or the latter. First, we need to know how Dirac spinors transform
under P. Again, I won't derive it, but simply quote the result (problem 7.12):'
(7.61)
It follows that
so (ifi1/f) is invariantunder P
out of 1/1:
-
(7.62)
it's a 'true' scalar. But we can also make a pseudoscalar
(7.63)
where
(7.64)
• n.., Jig>! in Equ:Uion 7.61 is pure con'<ention; -rOt would do just u weU.
7.3 BilineorCOlIQrionts
['11 let you check that it s
i Lorentz invariant (problem 7.14). As for its behavior
under parity
(7.65)
(I used the fact that (y O)2 = 1 in the last step.) Now, the yO is on the wrong side' of
the yS, but we can 'pull it through' by noting that it anticommutes with y I , y2, and
yl (Equation 7.15) and commuto" (of course) with itself (y l yO = _ yOyl, y2yO =
_ yO yl , y l yO = _yO yl, yO yO = yOyo ), so
'
By the same token, y5 anticommutes with all the other y matrices:
(7.66)
At any rate
(7.67)
so it's a pstudoscalar.
All told, there are 16 products ofthe form "''''''j (taking one component from "'"
and one from "'), since i andj run from 1 to 4. These 16 products can be assembled
in various linear combinations to construct quantities with distinct transfonnation
behavior, as follows:
(1)
(21
(3)
(4)
(5)
�'" = scalar
liiy s", = pseudoscalar
�y"'" = vector
li/y"yS", = pseudovector
Vier"·"' :: antisymmetric tensor
(one component)
(one component)
(four components)
(four components)
(six components)
(7.68)
where
".
o:r
!!!!
i
-(y" y" - y.y")
2
(7.69)
This gives 16 terms, so it's all we can hope to make. You cannot, for example,
construct a symmetric tensor bilinear in "'. and "" and f
i you're looking for a vector,
Viy"'" is the only candidate." (Another way to think of it is this: 1, y5, y", y" y5,
and o:r"" constitute a 'basis' for the space ofall 4 x 4 matrices; an}/ 4 x 4 matrix can
be written as a linear combination ofthese 16. In particular, ifyou ever encounter a
• Notice thlt jy0.;- "" .;-tyOyOt "" ttoj, 50 .;-toj is lctu.ally the zeroth component of a
four·veaor. "Th.at's why th nornulization con".,ntion (Equation 7.41), which no doubt IooW
peculiu It the time, is actually very sensible. By normalizing "1,, to the =oth component of
the four·vector pI', we obtain a rdativistically 'natural' convention (Stt Problem 7.16).
1237
2381 7 Q�onjllm ElulrotiynomiG!
product of five y matrices, say. you may be sure that it can be reduced to a product
ofno more than two.)
Pause a moment to admire the n
i genious notation in Equation 7.68. The tensorial
character ofthe bili�arrovariants, and even their behavior under parity. is indicated
at a glance: Vlyl'1/I looks like a four-vector, and it is a four-v«tor. But yl' by itselfis
certainly nota four-v«tor; it's a co!l«tion offour fixed matrices (Equation 7.17); they
don't change when you go to a different inertial system - it's 1/1 that changes, and in
just such a way as to give the whole 'sandwich' the tensorial taste of the jam inside.
7.'
The Photon
In classical el«trodynamics, the el«tric and magnetic fields (E and B) produced by
a charge density p and a current density J are determined by Maxwell's equations:'
I
aB
(il) V x E + �
=0
c a,
(i)
V · E = 4np
(iii)
(iv)
the 'field strength tensor'. P'":
)
V·B=O
I aE
4n
V x B - -- = -J
c at
c
I
(7.70)
In relativistic notation, E and B together form an antisymmetric s«ond-rank tensor,
-E,
0
B,
-B
,
(that is, pOl = -
E", Fll =
]I' = (cp,1l
- E,
-
- B,
0
B,
- E'
B,
-B,
0
(7.71)
B%. etc.). while p and , constitute a four-v«tor:
(7.72)
The inhomogeneous Maxwell equations. (i) and (iv) in Equation 7.70. can be written
more neatly in tensor notation (Problem 7.20)
(7.73)
From the antisymmetry of P" (P''' = -P'") it follows (Problem 7.20) that 1" is
divergenceless:
a,.]1' = 0
(7.74)
Or, in three-vector notation. 'if . J = aplat; this is the 'continuity equation',
expressing local conselVation ofcharge (Problem 7.21).
-
, This section presupposes some familiarity with dassi<;al tlectrodynamics: it is designed 10 make
the description of photoM in quantum electrodynamics more pLousible. As always. I use GlUS­
sian cgs units.
As for the homogeneous Maxwell equations. (iii) in Equation 7.70 is equivalent
7.4 Th. Plwto"
to the statement that B can be written as the curl of a ve<tor potential. A:
(7.7S)
With this. (ii) becomes
Vx
( I )
,
aA
E + --
"
=0
(7.76)
which s
i equivalent to the statement that E + (ljc)(aAjat) can be written as the
gradient ofa scalar potential. V:
I 'A
E = -V V - - ,
"
(7.77)
In relativistic notation. Equations 7.75 and 7.77 become
(7.78)
where
AI' = (V.A)
(7.79)
In terms ofthis four·vector potentia!. the inhomogeneous Maxwell Equations (7.73)
read:
(7.80)
In classical electrodynamics the jitlds are the phYSical entities; the potentials
are simply useful mathematical constructs. The virtue of the potential formulation
is that it automaticaUy takes care of the homogeneous Maxwell equations: given
Equations 7.75 and 7.77. (H) and (iii) in Equation 7.70 follow immediately. no
matter what V and A might be. This leaves us only the inhomogeneous Equation
(7.80) to worry about. The deftct of the potential formulation is that V and A
are not uniquely determined. Indeed. it is clear from Equation 7.78 that new
potentials
(7.81)
(where ). is any function of position and time) would do just as well. since
'
a"A" - a"AI" = al'A" - a"AI'. Such a change of potentials. which has no effect
on the fields. is called a gauge transformation. We can exploit this gauge freedom to
impose an extra constraint on the potential ]3]:
al'AI' = 0
(7.82)
1 239
2<10
I 7 QIIOt1ll1m EltClrodynomics
This is called the Lorentz condition; with it Maxwell's equations (7.80) simplify still
further:
DAI' = 411" J"
,
(7.83)
Here
(7.84)
is the relativistic extension of the Laplacian (Vz); it is called the d'Altmbertian.
Even the Lorentz condition, however, does not uniquely spedf)r AI'. Further
gauge transformations are possible, without disturbing Equation 7.82, provided
that the gauge function'\' satisfies the wave equation:
0,\. = 0
(7.85)
Unfortunately, there is no clean way to eliminate the residual ambiguity in AI',
and one must choose either to live with the indeterminacy, which means carrying
along spurious degrees of freedom, or to impose an additional constraint, which
spoi
l s the manifest Lorentz covariance of the theory. Both approaches have been
used in formulating QED; we shall follow the latter CourSE'. In empty space, where
l" = 0, we pick (see Problem 7.22)
AO = 0
(7.86)
The Lorentz condition then reads
V ·A=O
(7.87)
ponent (Ao) for special treatment, it ties us down to a particular inertial system
This choice (the Cou!.omb gauge) is attractively simple, but by sele<:ting one com·
(or rather, it obliges us to perform a gauge transformation in conjunction with
every Lorentz transformation, in order to restore the Coulomb gauge condition).
In practice, this is very seldom a problem, but it is aesthetically displeasing.
In QED, AI' becomes the wave function of the photon. Thefru photon satisfies
Equation 7.83 withl" = 0,
DAI' = O
(7.88)
which we recognize in this context as the K1ein�Gordon Equation (7.9) for a
massless particle. As in the case of the Dirac equation, we look for plane-wave
solutions with four-momentum p = (Ele, p) :
A"(x)
:=
at-(i/�)l'""�I'(P)
(7.89)
Here �I' is the polarization vutor � it characterizes the spin ofthe photon - and a
s
i a normalization factor. Substituting Equation 7.89 into Equation 7.88, we obtain
7.S The FeynmQn RlJle'forQED
a constraint on pI';
P"PII> e::o 0,
or
E
e::o
Iplc
(7.90)
f"
which is as it should be for a massless particle.
Meanwhile,
has four components, but they are not all independent. lbe
Lorentz condition IEquation 7.82) requires that
(7.91)
In the Coulomb gauge, moreover,
(7.92)
which is to saythat the polarization three·vector (f) is perpendicular to the direction
of propagation; we say that a free photon is lransverstly polarized.' Now, there are
two linearly n
i dependent three-vectors perpendicular to p; for example, if p points
in the z direction, we might choose
f(ll = (1.0,0),
f(ZI
=
(0, 1.0)
(7.93)
Instead of fo�r independent solutions for a given momentum (too many, for a
particle ofspin I), weare left with only two.lbat sounds like toofew - shouldn't the
photon have three spin states? The answer is no: a massive particle of spin 5 admits
25 + I different spin orientations, but a massllss particle has only two, regardless of
its spin (except for 5 = 0, which has only one). Along its direction of motion it can
only have m, e::o +50r m, = -5; its helicity, n
i other words, can only be +1 or _l.t
7'>
The Feynman Rules for QED
In Section 7.2 we found thatfreeelectrons andpositrons ofmomentump = (Ele, p),
with E = Jm1c4 + picl, are represented by the wave functionsf
• This corresponds to me facl WI electromag.
t Photon states with m, = ±! correspond
ndie waves are transverse.
to right· and left·drcu�r polarization;
itl2ll/J2. Notice that it was
1M: respective polariuolion "«lOTS are
f ", ., 'f(fll! ±
by specifying a �rticular go,uge that we elim·
ilUtw the nonphysical (m, = 0) solution. If
which we avoid imposing the Coulomb gauge
we �re to follow a 'covariant" approach. in
condition. longitudinal free photons would
be pre...ni n
i the th�ry. But the... 'ghosts'
decouple from everything el.... ..nd they do
not affect the lilUl results.
� For ref�fKe, I begin with a sllmmary of the
es...nti..l results from earlier $e<:tions. I speak
of'doxtrons' and 'positrons'. but they cOllld
as �n be 1<- ..nd 1<+. or r - and r+, or (with
the appropriate deemc charges) quarks and
antiquarks _ in ShOTt, an� poinl charges of
spin t,
12'41
2421
7 Qua07lum [/eClrodv"omics
Electrons
Positrons
(7.94)
where s = 1 , 2 forthe two spin states. The spinors ul') and vl') satisfY the momentum
space Dirac equation(s):
(yl'pl' + melv = 0
(7.95)
(7.%)
They are orthogonal,
(7.97)
normalized,
uu = 2me
liv = -2me
(7.98)
and complete, in the sense that
L ulslul<l "" (y"p" + me)
...1.2
L vl<lijl<l "" (y"PI' - me)
...1.2
(7.99)
(see Problem7.24). A convenient explicit set (Ulll, IPI , vlll, (121) is given in Equations
7.46 and 7.47. Ordinarily, we'll be averaging over electron and positron spins, and
in that case it doesn't matter that these are not pure spin up and spin down - all
we really need is completeness. For the occasional problem in which the spins
are specified, we must, of course, use the spinors appropriate to the case at
hand.
Meanwhile, a free photon of momentum P = (E/e, p), with E = Iple, is repre­
sented by the wave function
Photons
A,,(x) = ae-I;lhIP'�t' )
t
(7.100)
where s = 1, 2 for the two spin states (polarizations). The polarization vectors
satisfy the momentum space Lorentz condition:
t'�<l
They are orthogonal, in the sense that
(7.102)
7.S The Feynman Rufesfor QED
Fig. 7.1 A generic QED di�gram, with exlern�1 lines I�beled. (Intern�1 lines nal shawn.)
and normali:zed
(7.103)
In the Coulomb gauge
�o = o, ( ' p = O
(7.104)
and the polarization three-vectors obey the compltkness rtlation (Problem 7.25)
L f)')f},j * = $ij - P;Pj
(7.105)
...1.2
(�(I), �Ilj) is given in Equation 7.93.
To calculate the amplitude, ./Ii, associated with a particular Feynman diagram,
proceed as follows:
A convenient explicit pair
Feynman Rules
1. Notation: To each txtl:malline associate a momentum
PI, P2, . • . ,pn, and draw an arrow next to the line, n
i dicating
the positive direction (forward in time).' To each imanalline
associate a momentum ql, qz, . . .; again draw an arrow next
to the line n
i dicating the positive direction (arbitrari
ly
assigned). See Figure 7.1.
2. Extemallines: Extemallines contribute factors as follows:
Incoming( _) , II
Outgoing(_) , il
Positrons :
Photon>: .
h>C(lming(_) : ;:;
OUlgoing(_) : t'
Incoming("""') : .�
Oulgoing(-) : ,�•
• For � fermion. of course, there will al",ady be an arrow on the line. telling uS whether it is an
electron or a pc»itron. The two arroWS have nothing to do with one another: they may or may
not point n
i the pme direction.
1 241
2«
I 7 QIJQMlllm fl�"rodyMami"
3. Vertexfactors: Each vertex contributes a factor
The dimensionless coupling constant g. is related to the
charge ofthe el&tron: g. :: e..,l47Cl1fi :: ../
4Jta..'
4. Propagf.ltors: Each internal line contributes a factor as
follows:
EleClrons and positrons:
Pholons:
S. Conservation ofe"-",rgy and momenJum: For each vertex, write
a delta function of the form
vertex (if an arrow leads outward, then I: is minus the
where the I:'s are the three four-momenta coming into the
four·momentum of that line).
6. Inlegra� over jn�mal momenta: For each internal
momentum q, write a factor
and integrate.
7. Caned Ihe deltafonction: The result will include a factor
corresponding to overall energy-momentum conservation.
Cancel this factor, and multiply by i; what remains is A.
[t is critically m
i portant that the pieces be assembled in the correct order ­
otherwise the matrix multiplications will be gibberish. The safest procedure is
• In HeaYisidr-L<>rrntz units. with Ii and , $tl
eqUll 10 1. g. is Ihe 'ha� of the poSitron,
writing the Feynman ru\rs for QED I assume
and hrn(e e. written 'e' in mosl �ts. In
we are dealing witlt elec!rons and positrons.
trouble o�r Unils i5 10 express all results in
P"n� (a5 oppo:s...! to tlte �ntipartide). For
elec.trons, q = -�, bUI for 'up' quuh, »y,
the. book I u$t G�ussian units, and k�p all
factors of Ii and c. Thr rasirs! way 10 avoid
lerms of thr dimensionless numbrr a. In
In grn<ral, tlte QED coupling constanl is
-q�, wllere q is the charge of tlte
q = l�
7.6 EKamples
to track tach fermion lint backward through tht diagram. Start (for example) with
an outgoing electron line, and follow the arrow (the one on the line) back until
it emerges, either as an incoming electron or as an outgoing positron, writing
down the various line factors, vertex factors, and propagators, left to right, as you
encounter them. Each fermion line produces a 'sandwich', of the form adjoint
spinor, 4 x 4 matrix, spinor (row · matrix · column number). Meanwhile, each
vertex carries a contravariant ve<:tor index (IL, IJ, A, . . .), which contracts with the
covariant index of the associated photon line or propagator. (Don't worry: all of
this will make much better sense when we work some examples, but I wanted to
prescribe the ritual, for future reference.)
As before, the idea is to draw all the diagrams contributing to the process in
question (up to the desired order), calculate the amplitude (A) for each one, and
add them up 10 get the total amplitude, which is then n
i serted into the appropriate
Golden Rule for the decay rate or the scattering cross section, as the case may be.
There's one new twist that occasionally arises: the antisymmetrization of fermion
wave functions requires that we insert a minus sign in combining amplitudes
that differ only in the interchange of two identical external fermions. It doesn't
mailer which diagram you associate the minus sign with, since the total will
be squared eventually anyway; but there must be a rtIative minus sign between
them:
8. Anlisymmtlrization: Include a minus sign between diagrams
that differ only in the interchange oftwo incoming (or
outgoing) electrons (or positrons), or ofan incoming electron
with an outgoing positron (or vice versa).
�
We are now in a position to reproduce many ofthe classic calculations in quantum
electrodynamics. Just so you don't get lost in the details, let me begin by giving
you a catalog of the most important processes (fable 7.1). The simplest case
s
i electron-muon scattering, for here only one diagram contributes in second
order.'
, It �sn't havr to br an eand a 1'. ofcou�.
Any spin. ! point charg<:$ would do (. and
T. for instance. Or p. and T. or eleetron and
quark, etc.). as long as you put in the cor·
reet mu�s and charges. As a matto:r of fact
most books u� electron-proton scattering as
the canonical ...ample, but that is acrually a
rother inappropriate choice. since tbe proton
is a composite strocrure. not a point particle.
Still, to the extent that the internal structure
of tbe proton can be ignored. it is not a wd
approximation (rather like lIeating the sun as
a point mns in the the<>ry of the oola, sys·
tem). If the 'muon' is much beavier than the
'electron', we have MOl: ""ll<ring: if, more­
over, the 'electron' is non'elativistk, we get
R..Wiford ""tiering, for which QED repro.
duces precisdy the classical formula (Example
6.4).
I
us
2<46 1
7 Q�ont�m E/edrociynQmi",
Tai)le 7.1 (at./og of b.sic quantum dectrodynamic processes.
Second--order processes
EIo.stic
x
XX
X >-<
XX
I
Ele.:tron-muon scattering(e + IJ. ---> e + IJ.I
(Mott scattering(M » m) � Rutherford sottering(1I « c))
Ele.:tron-ele.:tron scattering(.- +.- ...... .
I
(Meller scattering}
I
(Bhabha scattering)
- + e-)
Electron-positron scattering(r
+ .+ ..... e- + .+ )
/"daslic
xx
X X
{pairannihllarion(.- + e+ ---> )' + 1'1
{pair production(1' + I' ...... r
+ e+)
Most imporbnt third>order process
{ � Anomalous magnetic moment of electron
Example 7. 1 Electron-Muon Scotten"ng Walking 'backward' along each fermion
line (Figure 7.2), and applying the Feynman rules as we go:
(21r)4
f
ru'>l){Pl)(ig,y")�I'll(pl)J
�
-
.
rul"I(P4)(ig,y")ul>21<P2)J
x ,s4{P1 - pJ - q),s4(f2 + q - P4) d4q
Notice how the space·lime indices on the photon propagator contract with those
of the vertex factors at either end of the photon line, carrying out the (trivial) q
integration. and dropping the overall delta function, we find
(7.106)
In spite of its complicated appearance. with four spinors and eight y matrices.
this is just a numbu, which you can work out once the spins are spedfied (see
Problem 7.26). r�.:
7.6 Examples
,
�
,
"»
ql
�
�
Fig. 7.1. Electron-muon scattering.
,
,
Fig. 7.3 'Twisted· diagram for
electron-electron scattering.
Example 7.Z Electron-Electron Scattering In this case there is a second diagram.
n
i which the electron that emerges with momentum P1 and spin Sl comes from
the P1,51 electron. instead of the Pl.Sl electron (Figure 7.3). We can obtain
this amplitude from Equation 7.106 simply by the replacement P1.S1 ...... p•• 54.
According to Rule 8. the two diagrams are to be subtracted, so the total ampli.
tude is
A=­
+
�,
�
,
p,)
z (1i(3)y" u(1)J!u(4)y" u(2)]
,
z [U(4)Y" u(1)J[u(3)y" u(2)J
(PI p.)
(7.107)
�
(Note the transparent shorthand I have adopted to label the spinors.)
IIltl
Example 7.3 Electron-Positron Scattering Again. there are two diagrams.' The first
is similar to the electron-muondiagram (Figure 7.4):
f
(211")4 [ii(3)(ig.y")u(l)]
-�" [v(2)(ig.yV)v(4)]
4
x <'J·(P1 -Pl - q)<'J· (Pz + q -p4) d q
Notice that 'proceeding backwards' alongan antiparticle line means workingfoTWl1rd
in time; the order is always adjoint/matrix/spinor. The amplitude for this diagram
• The fact that there are IwO diagrams for
ekctron-dectron and electron-positron
scattering. but only �"" for electron-muon
scattering. would appear offhand to be in·
consistent with the classical limit. After .U.
Coulomb's law says that the force of attraction
or repulsion ]",tween two particles depends
only on their charges. nOI on whether they
hap!",n
]", identical (or antiparticks of
one another): in the nonrelalivistic limit.
then. we should get the same answer whether
10
we u� the electron-muon formula or the
ekclron-ekctron formula. The <>mplitudts. il
is true. are no: the same. but the cross section
formula (Equation 6.3-4-) carries a ractor of
S. which is for electron-electron scat·
�ring and I for electron-muon scattering.
For electron_positron scattering. S = I. but
the second amp�tude (Equ�tion 7.109) i s
sm�ner th�n th� first (Equation 7.}08) by a
factor
SO only..l, contributes. in the
nonrelativistic limit.
t
(\I/e)l.
1 20
248 1
7 Quotllum fJecwdvtlomics
,
Fig. 7.S Second diagram contributing to
electron-positron sc�nering.
Fig. 7.4 Electron-positron sc�nering.
is thus
Ai = -
,
(PI
go
P3)
2[ii(3)Y"u(1)J[V(2)Yl'u (4)j
(7.tOS)
The other diagram represents virtual annihilation of the electron and positron.
followed by pair production (Figure 7.5):
(2JT)·
f
�
[ii(3}(ig,yl')V(4)]
" [v(2)(ig.yV}u(I )]
x 8 (q - Pl - p,)6'(Pt + P2 - q}d q
�
'
The amplitude for this diagram is therefore
(7.109)
Now. should we add these amplitudes. or subtract them? Interchanging the incom·
ing positron and the outgoing electron in the second diagram (Figure 7.S). and
then redrawing it in a more customary configuration
we recover the first diagram (Figure 7.4). According to Rule 8, then. we need a
minus sign:
..it
=
-
+
(PI
.;
p))2
,
[ii(3)Y"u(1)][u(2)y" v(4)]
2[ii(3)y"v(4)1[V(2)y"u(I ))
(PI + Pi)
go
(7.110)
m
7.7 Casimir's Trick
Fig. 7.6 Compton sCinering.
Fig. 7.7 S<!(ond diagram for Compton seaner·
mg.
_
Example 7.4 Campton Scattering For an example n
i volving the electron propagator
and photon polarization, consider Compton scattering, y + e
y + t. Again there
are two diagrams, but they do "at differ by the n
i terchange of fermions, and the
amplitudes add. The first diagram (Figure 7.5) yields
(211")4
/ [U(4)(ig.y") �
( f$ m�) (ig.yV)U(l)] E.(3)'
+
(,,(2)
x �4(P1
_
pJ
_
q)�.(pJ. + q P4) d4q
_
Notice that the space.time index on each photon polarization vector is contracted
with the index ofthe y matrix at the vertexwhere the photon wascreatedorabsorbed.
Notice also how the electron propagator fits in as we work our way backward along
the electron line I have introduced here the very convenient 'slash' notation:
.
(7.111)
Evidently, the amplitude associatedwith Figure 7.6 is·
(7.112)
Meanwhile, the second diagram (Figure 7.7) yields
.;
=
:;- [ii ( 4)/( 3)' i.ll +12 + me)/(2)u{l)]
-c:,,,
)' -,-'
m"
(pC"
.,1{2 "
p,\+-:
�
(7.113)
and the total amplitude i s .At = -"'\ + -"'1' R
7.7
Casimir's Trick
In some experiments the incoming and outgoing electron (or positron) spins are
specified, and the photon polarizations are given. If so, the next thing to do is to
1249
2S0
I 7 QI/(lntl'm f/ectrodyn(lm;Cl
insert the appropriate spinors and polarization vectors into the expression for 4,
and compute 1411 - the quantity we actually need, to determine cross sections
and lifetimes. More often, however, we are not n
i terested n
i the spins. A typiCilI
experiment starts out with a beam of particles whose orientations are random. and
simply counts the numbu ofparticles scattered in a given dire<:tion. In this case the
relevantcross section is the av(rage over allinitial spin configurations, Sj, andthe sum
over all final spin configurations, sf. In principle, we could compute 14(5; -+ sf)l
l
for every possible combination, and then do the summing and averaging:
l
041 ) ;:: average over initial spins, sum over final spins,
ofl4(5; -+ sfl ll
(7.114)
In practice, it is much easier to compute {[41l1 directly, without ever evaluating
the individual amplitudes.
Consider, for instance, the electron-muon scatteringamplirude (Equation 7.106).
Squaring, we have
1412 =
(PI - PJ)
t
4
[U(J)y" u(I)][U(4)y" u(1)][U(3)y"u(I))"[U(4)You(2))"
(7.115)
(I use v for the second contraction, since /l has been preempted.) A glance at the
first and third 'sandwiches' (or the second and fourth) reveals that we must handle
quantities of the generic form
(7.116)
where a and b stand for the appropriate spins and momenta. and r\, and
rl are 4 x 4 matrices. All the other processes described in Section 7.6 - Moller,
Bhabha, and Compton scattering, as well as pair production and annihilation - lead
to expressions with similar structure. To begin with, we evaluate the complex
conjugate (which is the same as the Hennitian conjugate, since the quantity in
square brackets is a 1 x 1 'matrix'):
(7.117)
(7.118)
where'
(7.119)
t
• O�"'" that the overb>.r nQW strve$ two different functions. On a spinor it denotes th� adjoillt:
_ ttyO (Equation 75S); on a 4 x 4 ""',m it ddines a new matrix: f _ yOrt yO.
7.7 Co,imir's Tri�k
G = [ii(ujrJu(b)J[Ulb)f\u(u)J
(7.110)
We are now ready to sum over the spin orientations of particle b. Using the
completeness relation (Equation 7.99), we have
I
)
L G :::: ii(ujrJ L U('.J {P�)uI'.J {P�) r2U(U)
b.pin.
.._1.2
u Q
:::: u(U)rJ(,Iib + mbe)r2u(U) :::: (u) u(a)
(7.121)
where Q is a temporary shorthand for the 4 x 4 matrix
(7.112)
..
Next, we do the same for particle a:
L L G = L ij:I..J{P4)QU( J {p.)
4 .pin.
b .pi...
.._1.2
Or, writing out the matrix multiplication explicitly;·
z;;, It "'·I�.110J"'·'�.IJ It 0J I.E "" '�'I''''�'I
=
•
•
= L QJj(,li4 + m�e)p :::: L[Q(,Ii4 + rnA];, = Tr[Q(,Ii� + m.e))
)
,
(7.113)
where 'Tr' denotes the trace ofthe matrix (the sum of its diagonal elements):
Tr(A) .. LA�
(7.114)
Conclusion:
L [ii(ajrJ u(b))[ii(Ujr2U(b))" :::: Tr[r1 (,Ii� + m�c)r21!. + m4c)]
.11
.pin.
-
(7.115)
This may not look like much ofa simplification, but it is actually huge. Notice that
i
there are no spinoTS left once we do the summation over spins, all that remains s
to multiply matrices and take the trace. This is sometimes called 'Casimir's trick',
i either u (in
since Casimir was apparently the first one to use it
Incidentally, f
[4).
..
• This is r�ncy footwork, so w�tch closely. You Cln't me" with the ordering or two spinors, but
their c.omp"".lIIs �re just num"'rs 'MY Cln be written either WlY' Ui"j ":lUI· In the second
=
step we recognize this product as theji element or the m<llrix u (note the unusulirmotrix mul­
tiplication �re, column time. row: 4 x limes 1 x 4 produces 4 x 4).
1
1 251
252
1 7 Qua�wm flearodv�omics
Equation 7.115) is replaced by a t/, the corresponding mass on the right.hand side
switches sign (see Problem 7.28).
E.rcomple 7.5
and hence J\ = yOy "tyO = y. (Problem 7.29). Applying Casimir's trick twice, we
In the case of electron-muon scattering (Equation 7.115), r2 = y",
find
0.£12) =
•
4(PJ
&
Pl)
4 Tr[Y"V'l + mc)Y V'l + me)!
·
(7.126)
X TrjY" V'2 + Me)Y. V'4 + Me)!
where m s
i the mass of the electron and M is the mass of the muon. The factor
of � is included be.::ause we want the average over the initial spins; since there are
two particles, each with two possible orientations, the average is a quarter of the
sum.
_
Casimir's trick reduces everything to an exercise in calculating the trace of
some complicated product of Y matrices. This algebra is facilitated by a number
of theorems, which I shall now list (I'll leave the proofs to you - see Problems
7.31-7.34). First of all, I should mention three facts about traces in general: if A
and B are any two matrices, and a is any number
1.
Tr(A + B) = Tr(A) + Tr(B)
2.
Tr(aA) = a Tr(A)
3.
Tr(AB) = Tr(BA)
It follows from number 3 that Tr{ABq = Tr(CAB) = Tr{BCA), but this is not
equal, in general, to the trace of the matrices taken in the other order: Tr(ACB)
=
Tr(BAC) = Tr{CBA). You can 'peel' matrices ofT the back end of a product and
move them around to the front, but you must preserve the ordering. It s
i useful to
note that
4.
g,..g"" = 4
andto recal! the fundamental anticommutation relation for the Y matrices (together
with an associated rule for 'slash' products):
5'. # + � = 2a . b
From these there follows a sequence of'contraction theorems':
6. y"y"
7. y"y"y"
=
4
- 2y ·
7'. y"ly"
- 2;
7,7 CQsimir's TriG!<
4g"l
y" y"yly"
9. YI, y"yly" y"
8.
_
8'.
2y" yl,y"
••
=
y,,#y"
=
y,,#{y"
4{a· b)
1253
- 21#1
and a collection of 'trace theorems':
10. The trace ofthe product of an odd number of gamma matrices is zero.
II. Tr{l)
4
12. Tr(y"y")
lJ Tr(y"y"yl,y,,)
= 4(g"VgM
= "'"'
g"l,g"" + g""g"l,)
_
12'. Tr(i.b')
4(a · b)
13'. T,,,JlIA
= 4(a · b c d - a c b d + a d b c)
.
.
.
·
Finally, since yS = iyOy l y2yl is the product ofan even number of y matrices, it
follows from Rule 10 that Tr (ySy") = Tr (ysY"yVyl,) = O. When yS is multiplied
by an even number of y '5, we find
14 Tr(ys)
=
.
15. Tr(ySy"YV)
!
0
16. Tr(ySy" y"yl, y� )
where'
€"""" '='
Example 7.6
-1
+1,
0,
0
4if""""
= 0
15'. Tr(yS,J)'l
16'. Tr(ysJiN�}
4i€"""" a"b" cl,d"
if /-iVAU is an even permutation of 0123,
if/-iVAU is an odd permutation.
any two indices are the same.
(7.117)
f
i
Evaluate the traces in electron-muon scattering (Equation 7,126):
Tr[Y " {f1 + mc)Y"{fJ + me)]
=
Tr(yl'"lIY·"11) + me [Tr{Y""l1 y ") + Tr(y"y ""Ill] + (mc)lTr(y"y V)
• By '"""n prrmutation' I mean an e�n num·
00 of n
i t"",h.:Ing'" nf two indic.,•. Thu.
f"''"'' "" -f'�"" = f'''''� = _(.lc�. and so
on - in nther words, ��."" s
i antisymm�·
ric in e.-ery �jt ofsupe=ripts. It might
�Olll
is mill'" 1; why
� stranW' that
"",k., it pJ", l � It's purely enn"""tinn.I,
of cou..e - rndently. wh""""r establish."J
nn!
the definition want."J (om to be plus l.
.ince thrtt spatial indices are raised. By th.,
way, if you are UsM to worlcing with the
three·dimeosioMl Levi-Civit. symbol f;;t
(Probl.,m 4.19), b., warn."J that although
an even permutation on wu indices (Or·
respond. to preservation of cyclic order
((\11 = fJU = flij). this is not the case forfo,,'
indices: �"."" = -�....� '" �....�" = -(g�.,.
and from th.t it follows that �oUJ = -I,
25'-, QlUmlllm f/earodynomics
7
Solution: According to Rule 10, the terms in square brackets are zero. The last term
can be evaluated using Rule 12, and the first by Rule 13:
Tr(Y",IY·,l) = (Plll(Pl)"Tr(y" yly "y" )
(PJ)l(Pl).,.4(g"lg"<' - g""g'" +g"",,")
= 4(pfpj' - g""(PI . Pl) + p)'plJ
=
Tr [y"(fl + mc)Y"(fl + me)]
= 4 lpfpj' + PlPi + "" [(mc)2 - (PI . PJ)]I
(7.128)
The second trace in Equation 7.126 is the same, with m -+ M, 1 -+ 2, 3 -+ 4, and
the Greekindices lowered. So
X Ip2"p." + P4"P2" + g". [(MC)2 - !P2 . P4)]J
s.: 4 [(PI ' P2)(PJ . p.) + (PI . P.)!P2 · PJ)
p. )
- (PI . pJ)(Mc)l - (Pl . P4)(mc)l + 2(mMIf)1]
= (P
•
(7. 129)
7.'
Cross Sections and lifetimes
We are now back on familiar turf. Having calculated IAll (or, where appropriate,
(IAll)), we simply plug it into the relevant cross section formula from Chapter 6:
Equation 6.38 in the general case, Equation 6.47 for two-body scattering in the CM,
or one of the equations from Problems 6.8, 6.9, or 6.10 in the lab frame.
E:.omple 7. 7 Motl and Rutheiford Scattering An electron (mass m) scatters off
a much heavier 'muon' (mass M » mI . Assuming that the recoil of M can be
neglected, find the differential scattering cross section in the lab frame (M at
rest).
Solution: According to Problem 6.8, the cross section is given by
7.8 C"", Swio", Q"d Jjf�tim�
•
Before
After
Fig. 7.3 Ele�tron snttering from a heavy target
Because the target is stationary, we have (Figure
7.8):
where
the incident (and scattered) electron energy, PI is the incident
momentum, and Pl is the scattered momentum (their magnitudes are equal,
and the angle between them is B so PI . P, = cos O . Thus
(PI) =
E is
Ip,l ;;;; Ipl,
p2 )
-p))2 = -Pt - pi + 22PI . Pl
-4plsin2 2 W/2)
_2pl(12 2
2
(P1 ' P,) ( E/e) - PI ' P, =pl+m c pi cosO m 2+ 2plsin (0/2)
(PI . Pl)(Pl . P4) (PI . P4)(P2 ' Pl) = {ME)2
(F2 . P4) = (Me)2
Putting this into Equation 7.129, we have
(7.130)
J
and therefore (recalling thatg.. = 41Ta)
(7.131)
This is the Mott formula. [t gives, good approximation, the differential cross
section for el
n p ton scattering. If the n
i cident electron is nonrelativis·
tic, so p2 « (me)l, Equation 7.Bl red ces to the Rutherford formula (compare
(PI - PI)2 = -(PI
cosO) =
=
=
=
_
=
ectro
-
to
ro
u
Example 6.4):
(7.132)
...
What about decays? Actually there no such thing in pure QED, for if a single
fermion goes in, that same fermion must eventually come out; a fennion line
,
is
,
1 255
2561 7 Quantum EleClroifrnomics
Fig. 7.9 Two ,ontributions to pair annihilation.
cannot simply terminate within a diagram, nor is there any mechanism in QED for
converting one fermion (say, a muon) into another (such as an electron). To be sure,
there exist electromagneticdecays of wmpositt particles, for example, 11"0 --+ )' + y;
but the electromagnetic component in this process is nothing but quark-antiquark
pair annihilation, q + q --+ )' + y. It is really a scatttring event, in which the two
colliding particles happen to be n
i a bound state.
The cleanest example of such a process is the decay of positronium: e+ + e- --+
y + y, which we consider in the following example. We'll do the analysis in the
positronium rest frame (which s
i to say, in the CM frame of the electron-positron
pair). The electron and positron are moving rather slowly - indeed, for purposes
of calculating the amplitude we shall assume they are at rest. On the other hand,
this s
i one of those cases in which we cannot average over initial spins, because the
composite system is either in the singlet configuration - spins antiparallel - or in
the triplet configuration - spins parallel - and the formula for the cross section
(and hence the lifetime) is quite different in the two cases.'
Example 1.8 PairAnnihiialiant Computetheamplitude, .A.fore+ + e- -+ y + y,
assuming that the electron and positron are at rest, and in the singlet spin
configuration.
Solution: Two diagrams contribute (Figure 7.9). The amplitudes are (for simplicity
I'll suppress the complex conjugate signs on the �'s):
...Hi =
.A2 =
•
(p,
(p,
.:
pJ)2
.:
P.)Z
2)/.ljIl -,1 + mc)/lu(l)
(7.133)
V(2)/lljll -,. + mc)/.u(l)
(7.134)
m2,2 ii(
mz,z
As a Jt»t!er of faCl. y<>u ca" do tllis p,articu·
lar problem by Casm
i r
i 's trid::, btnu.., of a
rather speci.11 drcumstano:e: the singlet sta�
can only decay to an Mil numoo of plio·
\(Ins (predominantly 11>'0) and the triplet to
an odd number (usually IIuu). So in caicul,u.
ing the matrix dement for .... + e- .... y + y .
we are ooaomalicaJl)' ..,leeting out the singlet
configuration �n if the triplet was included
t Worning: This is not an easy calcuJ..tion.
in the sum over spins. S� Problem 7.J,Q.
though �ry step is reasonably straightfor.
ward. You may prefer to skim it (or skip it
altogether). The final rf;Sult will be usW onc�
Or twice later on, but it is not necessary \(I
Jt»ster the de�ils at this stag�. (Howe�r, !
do think it is an illumn
i ating application of
the Feynman rules.)
7.8 Cron Sections and lifelimes
and they add
(7.135)
With the initial particles at rest, the photons come out back-to-back, and we may as
well choose the z axis to coincide with the direction of the first photon; then
PI = me(I,O,O,O),
p) = me(I,O,O, I},
P2 = me(l,O,O,O},
p. = me(1,0,0,-I)
(7.136)
and hence
(7.137)
The amplitudes simplify somewhat ifwe exploit Rule 5' from Section 7.7:
But fJ has only spatial components (in the Coulomb gauge), whereas PI is purely
temporal, so p! €J = 0, and hence
.
(7.138)
Similarly
but (pJ . fJ) = ° by virtue of the Lorenn condition (Equation 7.91), so
(7.139)
Therefore
But IJ'I - me)u(l) = ° (Equation 7.33), so
(7.140)
By the same token
(7.141)
1157
2Sa
I 7 Quantum Elearotiynam"Cl
Putting all this together, we find
(7.142)
Now
so the expression in square brackets (Equation 7.142) can be written as
(7.143)
But
(7.144)
and therefore
• . ,,
o
)( 0
-if · f:.
• .
)
•,
0
(7.145)
In Chapter 4 (problem 4.20) we encountered the useful theorem
(if · a)(if -b) = a · b + io" · (a x b)
(7.146)
It follows that
(7.147)
(which we could also have obtained from Rule 5'). and
(7.148)
where 1:
'"
(� �).
as before. Accordingly
(7.149)
So far I have said nothing about the spins ofthe electron and positron. Remember
that we are interested in the sil1glet state:
It! - HI/'"
7.8 Cross �Nions Qnd Lifelimel
Alingl.. == (AtH - Atd/../2
Symbolically
(7.150)
AH is obtained from Equation 7.149 with 'spin up' for the ele<tron (u(l) in
Equation 7.46)
(7.151)
and 'spin down' for the positron (\J(2) in Equation 7.47)
v(2) = J2ifiC(0 0 1 0)
(7.152)
Using these spinors, we find
ii(2)yOu(1)
ii(2)I:y Ju(l)
==
0
(7.153)
==
-2rmi
(7.154)
So
(7.155)
Meanwhile, for A. i we have
(7.156)
from which it follows that
(7.157)
Thus the amplitude for annihilation of a stationary e+e- pair n
i to two photons,
which emerge in the dire<tions ±z, is
..k"';ngl«
==
-2../2 ii(�J x f:4h
(7.158)
(I note in passing that since Ai. = -..k". t, the triplet configuration (t J. + J. t)/../2
gives zero, confirming our earlier observation thatthetwo.photon decay is forbidden
in that case.)
Finally, we must put in the appropriate photon polarization ve<tors. Recall that
for 'spin up' (m, = +1) we have (see footnote to Equation 7.94)
f:+ == -(l/../2)(U,O)
�_ = (1/../2)(I,-i,O)
(7.159)
whereas for 'spin down' (m, == -I)
(7.160)
1259
260
I 7 Quantum f!taradynamics
If the photon is traveling in the +z direction, these correspond to right and
left.{:ircular polarization, respectively. Since the z component of the total angular
momentum must be zero, the photon spins must be oppositely aligned: t! or ! t.
In the first case we have
(7.161)
In the second case 3 and 4 are interchanged:
f] x f� ",, - i�
(7.162)
Evidently we need the antisymmetric combination, (t.j. - H)/../2. which should
come as no surprise: this corresponds to a total spin of zero, just as it did when we
combined two particles of spin 1. Again, the amplitude is (At'l - Atl f)/h, only
Ihis time the arrows refer 10 photon polarization. Finally, then:
(7.163)
(I have restored the complex conjugation of the polarization vectors, suppressed
until now; this simply reverses the signs in Equation 7.161 and 7.162.) ruJl
That was a lot of work, for a modest.looking answer.' What can we do with it)
In the first place, we can calculate the lotal cross section for electron-positron
annihilation. In the eM frame, the differential cross section is (Equation 6.47)
(7.164)
Here
(7.165)
and, since the collision is nonrelativistic
lpd "" mv
(7.166)
where v is the n
i cident electron (or positron) speed.t Putting all this together, we
find
(7.167)
• Once you get used to it the eY�IUiltion of Feynman diagrams b«omes a ttdious and m�hankaJ
process, ar>d. there aist a number of computer prognms that will do the hard W(lrk for you. In
p;ortkular. Mathem.tica and Maple both support useful p;ockages [5).
t We used � = 0 in calculating At. but obviously we cannot do so h�. Is there an n
i consistency
in this� Not really. Think of it tllliI way: .A (and also £" El.lp,I, and Ipd) could be exp;onded in
powers of vlt. What we have done is to ca1cu!a� the leading term in each exp;onsion.
7.8 Cross $er;1iom and Life1imel
Since there is no angular dependence, the total cross section is 4.rr times this (6]:
a
�
( )
"
'"
"
m
'
(7.168)
Does it make sense that the cross section is inversely proportional to the incoming
velocity� Yes: the more slowly the electron and positron approach one another,
the more time there is for them to interact, and the greater is the likelihood of
annihilation.
Finally, we can calculate the lifetime of positronium, in the singlet state. This is
clearly related to the cross section for pair annihilation (Equation 7.168), but what
is the precise connection? Well, going back to Equation 6.13,
do-
dn
1 dN
= Z dn
we see that the total number of scattering events per unit time is equal to the
luminosity times the total cross section:
N
(7.169)
= .sea
If p is the number of incident particles per unit volume, and if they are traveling at
speed v, then the luminosity (Figure 7.1 0) is
(7.170)
Z : pv
For a single 'atom', the electron density is 11/'(0)1 2 , and N represents the probability
of a disintegration, per unit time - which is to say, the de<:ay rate. Thus
(7.171)
Now, in the ground state
2
11/'(0)1
= -;I (a�)'
Zit
(7.I72)
(Problem 5 23), so the lifetime of positronium is
.
1
21i
r
Cfsmc
r = - = --2 = 1.25 x lO-iO s
which is the result I quoted back in Chapter 5 (Equation 5.33) .
A
. . .. .. . . . . .
" ..
.. .
. . . ... ... ..
. ". . . . . . :
•
."
Fig. 7.10 The number of particles in the cylinder is pAvdl.
so the luminositjl (number per unit �rea Pf'r unit time) is pll.
(7.173)
1 261
2621 7 Quamum fi'CIrodynamics
7.'
Renormalization
In Section 7.6 we considered 'electron-muon' scattering, described in lowest order
by the diagram
and by the corresponding amplitude
(7.174)
with
(7.17S)
There are a number of fourth·order corrections, of which perhaps the most
interesting is 'vacuum polarization':
Here the virtual photon momentarily splits into an electron-positron pair, leading
(as we saw qualitatively in Chapter 2) to a modification in the effe<tive charge of
the electron. My purpose now is to indicate how this works out quantitatively.
The amplitude for this diagram is (Problem 7.42)
(7.176)
7.9 R,nornw/izQlion
Its inclusion amounts to a modification of the photon propagator:
(7.177)
where (comparing Equations 7.174 and 7.176):
(7.178)
Unfortunately, this integral is divergent. Naively, it should go like
J
Ikl } dlkl
:�:: f
=
Ikl dk = Ikll,
as Ikl --+ 00
(7.179)
{that is, it shouldbe 'quadratically divergent'}. In actual fact, because ofcancellations
in the algebra, it only goes like In Ikl (it is 'logarithmically divergent'). But never
mind - either way, it blows up. We encountered a similar problem in Chapter 6; it
seems to be characterisic
t of closed-loop diagrams in the Feynman calculus. Once
again, the strategy will be to absorb the infinities into 'renormalized' masses and
coupling constants.
The ntegra!
i
n
i Equation 7.178 carries two space-time indices; once we have
integrated over k, the only four-vector left s
i if", so 11'" must have the generic form
g".( ) + ql'q.{ ), where the parentheses contain some functions of ql. We WIite it
thus [71:
(7.180)
The second term contributes nothing to .4(, since the ql' contracts with yl' in
Equation 7.176, giving
while, from Equations 7.95 and 7.96,
and hence
(7.181)
i Equation 7.180 As for the first term,
So we can forget about the second term n
appropriate massaging of the integral (7.174) reduces it to the form (Problem
7.43)
I(q2) =
JL
12rr1
1f.<X>.. - !. z(I - z) [ - LZ(I - Z)] "'I
1
Z
dz
6
0
1
In
1
mlcl
(7.182)
2&4 1 7 QUQlltum flectrodytIQmics
The first integral cleanly isolates the logarithmic divergence. To handle it, we
temporarily impose a cutoff M (not be confused with the mass of the muon),
which we will send to infinity at the end ofthe calculation:
(7.183)
The second integral
f(,,) 6 [ z(1 - z) In{l + 1'<:(1 - zlldz
(
= _ � + � + 2 " - 2) jX + 4 tanh-! j "
(7.184)
"
"
3
"
x+4
is cumbersome but perfectly finite (Figure 7.11); the limiting expressions for large
and small x are
(x « I))
(7.185)
f(x) ;;: { l��
(x» 1)
Thu,
(7.186)
l(q'l � 1�' H�:) - f (;;,�)l
Notice that ql is negative, here: if the incident electron's three-momentum in the
is p, and the scattering angle is then (Problem 7.44)
'l =-4p sm -B
(7.187)
2
Thus _qllmlcl _ !.Illcl, and the limiting cases in Equation 7.185 correspond to
nonrelativistic and ultrarelativistic scattering, respectively.
to
=
9,
eM
2
..2
fix)
.
1
/
/
,
I
x
Fig. 7.11 Graph off(K) (Equ�tion 7.184). The solid line i�
the numeric�1 rt�utt; tht d�shed lint below it i� InK (which
approximates [(x) at large x); the straight line above it is
tIS (whi h approximates f(t) at small xl.
c
1.9 Rel'OrmQlizGlioll
The amplitude for electron-muon scattering, including vacuum polarization, is
therefore
J( = -g}[i{P
i l)Y�U(Pl)i
� {I
-
x [ii(P.)y·u!P2)]
� [ (�:) f (�[2 )]}
1
2
In
-
(7.188)
Now comes the critical step, in which we 'sop up' the infinity (contained for the
moment in the cutoff
M)
by introducing the 'renormalized' coupling constant
g) In
\-121T2
( M')
m2
(7.189)
Rewriting Equation 7.188 in terms of gR, we have
(7.190)
(Equation 7.188 is only valid to order s: anyway, so it doesn't matter whether we
i side the curly brackets.)
use g. or gR n
There are two important things to notice about this result:
1. The infinities are gone. There is no
M
in Equation 7.190. All
reference to the cutoffhas been absorbed into the coupling
constant. To be sure, everything is now written in terms of
gR, instead of g.. But that's all to the good: gR, not g., is what
we actually measure in the laboratory (in Heaviside-Lorentz
units it is the charge of the electron - or muon - and we
determine it experimentally as the coefficient of attraction or
repulsion between two such particles). If, in our theoretical
analysis, we look only at 'tree level' (lowest-order) diagrams,
we are led to suppose that the physical charge is the same as
the 'bare· coupling constant, g.. But as soon as we include
higher-{)rder effects we find that it is really gR, not g., that
corresponds to the measured electric charge. Does this mean
that our earlier results are all wrong? No. What it means is
that by naively interpreting g. as the physical electric charge
we were unwittingly taking into account the divergent part of
the higher-order diagrams.
2. There remains thefinite correction term, and here the
important thing to notice is that it deFl1ds on q2. We can
absorb this, too , into the coupling constant, but the 'constant'
i now a function ofq2; we call it a 'running' coupling
s
constant:
(7.191)
1 265
266 1
7 Quonlum fltclrQdynQm�
or, in terms of the fine structure 'constant' (g. = .J4Jra):
(7.191)
The effective charge of the electron (and the muon), then,
depends on the momentum transferred in the collision.
Higher momentum transfer means closer approach, so
another way of saying it is that the effective charge of each
particle depends on how far apart they are. This is a
consequence ofvacuum polarization. which 'screens' each
charge. We now have an explicit formula for what was, in
Chapter 2, a purely qualitative description. How come
Millikan and Rutherford, or even Coulomb, never noticed
this effect? If the electron's charge is not a constant. why
doesn't this foul up everything from electronics to chemistry?
The answer is that the variation is extremely slight, in
nonrelativistic situations. Even in a head-on collision at foe,
the correction term in Equation 7.192 is only about 6 x 10-6
(Problem 7.45). For most purposes, therefore, a(O) "'" 1:7 wi
ll
do just fine. Nevertheless, the second term in Equation 7.192
makes a detectable contribution to the lamb shift (8). and it
has been measured directly in inelastic (+e- scattering (9).
Moreover, we shall encounter the same problem n
i quantum
chromodynamics, where (because ofquarkconfinement) the
short·distance, relativistic regime is the case of interest.
I have concentratedon one particularfourth-order process (vacuum polarization),
but there are, ofcourse. several others. There are the 'ladder.diagrams':
These are finite and present no particular problems. But there are also three
divergent graphs:
xxx
(and of course three more in which the extra virtual photon couples to the
muon). The first two renormalize the electron's mass; the third modifies its
magnetic moment. In addition, all three, considered separately, contribute to the
7.9 R�"ormoJjlorjo"
renormalization of the electron's charge. Luckily, the latter contributions cancel
one another, so Equation 7.189 remains valid. {I say 'luckily', for these corrections
depend on the mass of the particle to which the virtual photon line attaches, and if
they did not cancel we would have a different renormalization for the muon than for
the electron. The Ward identity (the official name for this cancellation) guarantees
that renormalization preserves the equality of electric charges, irrespective of the
mass of the carrier).' And then . there are even higher·order diagrams, such as
,
, ...
These n
i troduce further tenns n
i Equation 7.192, of order (1'2,
(l'l, and so on, but I
won't pursue the matter here; the essential ideas are all on the table.
References
I Bjork�n, J. D. and Dn-II, S. D.
(1%4) Rtlativislil: Quonlum M�­
ckaniCl and R�laljvislic Quan·
tum Fitlds, McGraw HiIl. New
York.
2 For further discussion see Bjorken,
J- O. md Orell, S_ D. (1%4)
Relativistic Quantum Muhanics
and Rtlotivislic Quanlum Fidds.
McGraw·Hill, New York. Chapler 5.
or Seiden, A. (2005) Particle Physics:
A Comprekellj"" Illtroductioll, Ad·
dison Wesley, San Francisco, CA,
Section 2.14.2_
1 Jackson. J. D. (1999) Classical El«:·
trodjlllamiCl. 3rd edn. John Wiley
& Sons, New York. Sect. 6.3.
� Pais. A. (1986) Illward Bound,
Oxford, New York. p_ 375.
5 For a revi�w article on automatic
computation of Feynman dia·
grams see Harlander. R. and
Sleinnauser. M. (1999) Progress ill
Particlt: and N..au.r Phl"ics, 43, 167.
·
6 For more eleg,mt derivations
of Equation 7.168. see lauch,
J. M. and Rohrlich, F. (1975)
The Theol)' af PMtons alld E/u·
trOllS, 2nd edn, Springer·Verlag,
New York. Sect. 12·6, or: (a)
Sakurai, J_ ,. (1%7) Ad""nad Quall­
tum Mechallics, Addison-Wesley,
Reading. M,A., pp. 216 ff.
1 My notation follows thaI of
Hab.en. F. and Manin. A. O.
(1984) Quarks and l.eplOltl, John
Wiley & Sons, New York. Cnap.
7, and Bjork�n, J. D. and Dr�n,
S. D. (1964) Relativs
i ic Quant"m
Mtdw.ni<;s and Relativistic QUOII­
lum Fidds. McGr.lw-HiU, New
York. Cnap. 8. [ refer the reader
to these lexts, or to Sakurai, 1- 1(1967) Advanced Quolllum Mechan­
ics, Addison-Wesley. R��ding_ M.A.
pp. 216 ff, for further discussion,
t
• Of course. we can put � "'l1<1li bubble onto a photon line. just as we did with the tlearon in
Equation 7_176 BUI this will modify the electron and muon charge (and for tnal matter the tau
h«ause it is tlte lightest charged particle.
and the quarks) by the same amount. The electron insertion is the dominant correction. simply
1 261
2681 7 Quanlllm E/edrodynamicl
9 Levine, l. tt as (1997) ph}"i·
cal Rcvitw I.ellUs, 78, 424.
8 See, for example, Hab:en, F. and
Martin, A. D. (1984) Quorb WId Lep·
Ions, John Wiley & Sons, New York.
Secl 7.3,
Problems
7.1 Show that a4>/ax" is a covariont four·vector (ot> is a salar function of ", y, z, and I).
[Hint: First determine (from Equation 3.8) how conriant four-vectors transform; then
'
use (}4>/ax'" = (a4>I(}"")(}""I(}x" ) to find out how a4>I(}x" trmsforms.)
7.2 Show that Equation 7.17 satisfies Equation 7.a
7.1 Derive Equation 7.45, using Equations 7.43, 7.46, and 7.47
7.4 Show tha.t Il111 and ul�1 (Equation 7.46) ne orthogonal, in the sense that uilitull) = O.
�,show that VIL) and ull) are orthogonal. Are "II) and u11) orthogonal?
7.S Show that for ulll and ull) (Equation 7.46) the lower components (ua) are smaller
than the upper ones (uA), in the nonrelativistic limit. by a factor \lj�. [This simplifies
matters, when we are doing nomelativistic approximations; we think of u" as the 'big'
components and UI as the 1ittJe' components. (For Vii) and ulll the roles are reversed.)
In the relativistic limit, by contrast, u" and us are compa!1lble in size.)
7.6 Ifthe zaxis points along the directionofmotion,showtha.tll(1) (Equation 7.46) reduces to
ull)
=
and construct II(/), u(l), and v(/). Confirm that they aTe all eigenspinors ofS" and find
the eigenvalues.
7.7 ConslTuct the normalized spinors ul+) and "H representing an electron of momenrum
ofthe hdicil:)' operator lP ' I: with eigenvalues ± 1.
p with helicil:)' ± 1. That is, find the u's that satisfy Equation 7.49 and are eigenspinors
[
(� )
( ) ""IE",,+:;,:=��' I" l
SoIUlion : UI±I = A
where u =
(E + "'" I U
PO ± [PI
Px + iPy
'
and A2
=
2[p [c( l p l ± p�)
e
7.S The puryose oflhis problem is 10 dem<)nllralt lhat particles desaibaI by lh Dirac tqlllltion
carry 'intrin,;e' angular momenlum (S) in a<!dition to their orbital angular momenlum (L),
",,;Iher ofwhieh s
i separatdy umstlVtd, although lheir Slim is. II should be attempted only if
)'Ou are reasonablyfamiliar with qlllllll llm muhWlies.
(al Construct the Hamiltonian, H, for the Dirac equation. (Hint: Solve Equation 7.19 for
pOe. Solutio/!: H '" eyo(y . p me), where p _ (riji)V is the momentum ope!1ltor.J
(hI Find the commutator of H with the orbitll angular momenrum L . r x p. (Solulio/!:
(H, L) = -i/iL-yo(y x p)J Since (H, L) is not zero, L by itseJfis notconserved. Evidently
+
7.9 Renormalizafion
there is some olher form of �ngular momentum lurking here. Introduce the 'spin
angubr momentum', S. defined by Equation 751
(e) Find the commutator of H with the spin angular momentum, S .. (/i12)!:. [Solutio..:
[H,S[ = ili<:yo(y x pJl [t follows tlut the Iota! angular momentum, , = L+ S, is
conseIVed.
[d)Show that every bispinor is an eigenstate OfSl. with eigetlv;r.lue 1I1s(s+ I). and find
So What, then, is the spin of a particle described by the Dintc equation,
7.9 The charge conjugation operator (q takes a Dirac spinor 'It into the 'charge<onjugate'
spinor 'II,. given by
where y1 is the third Dirac 8"mma matrix. ISee Hal1.en and Martin 17). Sect 5.4.) Find
the charge<onjugates of ull) and "ll), and compare them with vll) and vl1l .
7.10 In going from Equation 7.18 10 Equation 7.19. we (arbi/r.l.rily) chose to worlc with the
i
factor contaning the minus sign. How would Section 7.2 be changed f
i we were to
replace Equation 7.19 by y"p + me = O?
"
7,11 Confirm the transformation role (Equation 7.52, with 753 and 7.54) for spinors. [Hittl:
we want it to carry solutions to the Dirac equation in the original fntme to solutions in
the primed frame:
il\y"a,,'It -mcy, = o
where y,' .. Sy, and
_
iIIy"a�y,' - me y,' = O
a
ax" a
ax"
a' - -- - --- - --,
" ax!"
ax!" ax" ax'"
•
[t follows that
The (inverse) Lorentz transformations tell us ax"lax"'. T:.>ke it from there.]
7.12 Derive the transformation role for parity, Equation 7.61, using the method in Prob­
lem 7,11.
7.Il [>1 Staning with Equation 7.53. calculate st $, and confino Equation 7.57.
[hI Show tlut Sf yO $ = yO.
7.14 Show that ty�1ft is invariant under the transfonoation 7.52.
7.IS Show tlut the adjoint spinors iil,ll �nd 1)11.1:) satisfy the equations
u(y"p" - me) = 0,
�(y"p" + me) = 0
IHim: Talce the transpose conju8"te of Equ�tions 7.49 and 7.50; multiply from the right
by yO. ;md show that (y")t yO = yOy".)
7.16 Show that the normaliution condition (Equation 7.43), expressed in tenos ofthe adjoint
spinors, becomes
uu = -�v = 2me
7.17 Show that ty"1ft is a four-vector, by confinoing that its components obey the Lorentz
transfonoations (Equation 3.g). Check that it transfonns as a (polar) vector under parity
1269
270
I 7 QuaMlu", Electrodyl1amjcs
(that s,
i the 'time' component is invariant, whereas the 'spatial' components switch
sign).
7.\8 Showthat the spinor representing an electron at rest (Equation 7.30) is an eigenstate of
the parity operator. P. What is its intrinsic parity� How about the positron? What f
i you
changed the sign convention in Equation 7.61 � Notice that whereas the al>soIut.: parity
of a spin0! p.uticle is in a sense arbitrary. the fact that patticles and antiparticles carry
opposiu parity s
i nol ubitnry.
7.19 lat Express y�y' as a linear combination of I, 1'1, y�. y� yS, and o�".
(btConstruct the matrices a lI, oll , and all (Equation 7.69), and relate them to 1: 1, 1:1,
and 1:] (Equation 7.SI).
7.20 la) Derive Equations 7.70 (i andiv) from Equation 7.73.
(bl Prove Equation 7.74. from Equation 7.73.
7.21 Show that the continuity equation (Equation 7.74) enforces conservation of charge. pf
7.22 Show that we are ;r.Iways free to pick AO = O. in free space. That is. given a potential A�
you don't see how to do this, look in any electrodynamics textbook.)
which does nol satisfy this constraint, find a gauge function A, consistent with Equation
7.8S. such that AO (in Equation 7.81) is zero.
7.23 Suppose we apply a gauge transfonnation (Equation 7.81) to the plane·wave potenti;r.l
(Equation 7.89). using as the gauge function
where I( is an arbitrary constant and p is the photon four·momentum.
lat Show that this A »tisfies Equation 7.8S.
fb) Showthat this gauge transformation has the effect ofmodifying l� . l� ... l� + KP".
(In particular, if we choose K = -lOIrfJ we obtain the Coulomb gauge polarization
vector, Equation 7.92) Uris observation leads to a beautifully simple test for the
gauge invariance ofQED results: the answer must be unchanged ifyou replace l� by
7.204 Using u(l), "Il) (Equation 7.46) and ViI), vll) Equation 7.-47). prove the completeness
f� + KP".
(
relations for spinors (7.99). [Not.:; UII is the 4 x 4 matrix defined by (uil)� • Uillj.)
7.25 Using til> and f(l) (Equation 7.93), confirm the complete�ss relation for photons
(Equation 7.10S).
7,2(; Ev;r.luate the amplitude for electron-muon scattering (Equation 7.1(6) n
i the eM
system, assuming the t and /J. approach one another along the z axis. repel. and return
back ;r.Iong the z axis. Assume the initial and final particles aU ve helidty +1. (AMni'I!r:
..A" = -lgJ)
ha
7.27 Derive the amplitudes (Equations 7.133 and 7.134) for pair annihiliotion, e+ + e- ...
y +y.
7.28 Work out the an;r.log to Casimir's trick (Equation 7.12S) for aMlip<lrliae!
L [ii(a)rlv(b))[V(a)r2v[b)]"
.n .pin.
and for the ·mixed· cases
L [ii(a)r\v(b))[ii(a)r2V(b))' and
.u .pin.
L [ii(a)r l u(b))[ii(a)r2u(b))*
an
spin<
7.2' (a) Show that yOy-1"yO = y", for v = 0, I, 2, and 3.
(b) Ifr s
i any product ofy matrices (r = y.YJ. . . . I'd show that f (Equation 7.119) is the
same product in reverse order, f = y
YJ.y
•
. .
•.
7.lO Use
Cu
imir's trick to obtain �n �xpression analogous to Equation 7.\16 for Compton
7.9 Renorma/jzatjan
suttering. Note that th�r� ar� four tenns her�:
7.31 (at Prov� trace theorems 1, 2, and 3. in Section 7.7.
[hI Prov� Equation
4.
5'.
(el Using the anticommut;otion relation S, prove
7.n (al Use th� anticommut;otion relation 5 to prove the contraction theorems 6. 7. 8, and
7';
9,
(bt From 7, prove
from 8, prove 8'; from
prove
and \3.
7.B (at Confinn the trace theorems
10. 11. 12,
9'.
9.
M From 12, prove 12'; from \3, prove 13'.
7.3� (aJ Prove theorems 14, 15, and 16.
(bJFrom
...
15. prove IS'; from 16, prove 16'.
.... ��
_ -6 6� (summation over !J.. v,;\. implied).
�....��"'t = -2(8� 6� -�; �n.
�. f
leI Find th�analogous fonnula for �
7.35 (aJ Show that (�
[bJ Show that �
...
�H"
(dJ Find the analogous formula for f� ·I."f.." .
:
ofthe mixed (co/contravariant) metric tensor 8� = g", = g,�.J
[Here
�� is the Kronecker delta: 1 if!J. = v, 0 otherwise. It an also be written in tenns
- y'
yl)
7.36 Evaluate the following traces:
(b)Tr[(I + me) (I + Me) (j + me) (I + Me)], where P is the four·momenrum ofa (real)
particle ofmass m and qis the four·momenrum ofa (real) particle ofma!s M. Express
('J Tr[y�y' (1
(I + yl) Yl.J
your �nswer in terms of m,
.
..
M. c. and (P' q).
7.37 Starting with Equation 7.107, determine the spin·averaged amplirude (analogous to
7.129) for eI�stk electron-electron scattering. Assume we're working �t high
�nergies, so that the mass ofth� electron an be g
i nored (i.e., set m
0). [Hinl: You
an read (lAIII) and (IAl(l) from Equation 7.119 For (AlAi) use the S<lme strategy
Equation
as Casimir's trick to get
.. +
sless
Then exploit the contraction theorems to evaluate the trace. Notice that for mas
pilrticles til<: conservation of momentum (PI + PI
PJ · P,.Pl · P I - PI p• . andpl P, = Pl · PJ.]
PI
po) implies that PI
.
. P2 =
.....
7.38 ( ) Starting with Equation 7.119, find the spin·averaged �mplitude for electron-muon
scattering in the eM frame, in the high·energy regime (m, M
0).
(b) Find the CM differential cross section for eleo::tron-muon sanering at high energy.
[A"'�"
(lie)' £ ( + c"" I' )]
Let E be the ele.::tron energy and 0 the sc.anering angle.
.
du
=
dQ
SIT
2£2
'
sln� I}/2
·
..... 0).
7.39 ( ) Using the result of Problem 7.37, determint the spin...averaged amplitude for
eleo::tron-electron sattering in the eM in the high energy regime 1m
112 1 7 QIIQn/llm f/ufrodynQm;CS
Fig. 7.11 Oeuy of the photon. y ..... ly - � prOC�$ forbid·
den by Furry's theorem (Problem 7.46).
(bl Find theeM differenti�1 cross section for electron-electron suttering �t high energy.
Compare your �nswer to Problem 7.38 (� footnote to Ex�mple 7.3).
1.40 Starting with Equation 7.1S8. calcul�te 1.4'12• �nduse Equation 7.105 losum over photon
polarizations. Check that the �nswer is consistent with Equation 7.1&3, and elIpiain why
this method gives the correct answer (nole th�t _ are now summing over aU photon
polarizations, whereas injaa the photons must be in the sn
i glet configuration).
7.41 Starting with Equation 7.149, akul�te (I.A'12) for .+ + e- ..... y + y, �nd use it to get
the differential cross section for pair annihilation. Compare Equation 7.1&7 (� footnote
before Example 7.8).
7.42 o,.,rive Equation 7.176 You'll need one last Feynman rule; for a closed fermion loop
include a factor -1 and take the tnce.
7.43 o,.,rive Equation 7.182 1Hinl: use the integral theorems in Appendix E of Sakurai 161.1
7.45 Evaluate the correction tenn in Equation 7.192 for the ase of� he;od·on collision in the
7.44 Derive Equation 7.187.
CM; �ssume the electron is traveling at �,. In the experiment 19J, the beam energies
were 57.8 GeV; what should the measured fine structure 'constant' have been? Look up
the �ctual result, �nd compare it with your prediction.
7."Ii Why an't the photon 'decay'. by the process y ..... y + Y (Figure 7.12)? Cakulate the
amplitude for this diagram. (This is an example of Furry·s Ihwn:m, which says that
any diagram containing a closd electron loop with �n odd number of vertices has an
amplitude ofzero.J
7.47 Surting with your answer to Problem 7.30, derive the Klein-Nishi"" fonnulo for
Compton scattering (in the rest frame of the target):
""
'0'
- . do
m1
(W')' [W'
«!
-
w
,
- + -' - sm 9
«!
«!
1
where «! and «!' are the frequeocies of the incident and scattered photons (Problem
3.27).
?robl.m, 48-50 pertQin tc Ih.
following model: /mugine fhal th. pMtcn, inslmd ift>eing " masslffl
vector (spin I) partitk. WU<ll M1IISSiVtS(;allir {spin OJ. SpuijimUy, SlIppost. 1M QEDvtrtexfiu:tcrW<'rt
(where 1 is !he 4 x 4 unil main>:), and Ike 'photon'propagalor w..r<
7.9 Renorma/izaliotl
-i
There is rnl phOlOn polarwnion vtClOr now. and kenu rnlfaclorfor alema/ pkolOn lines, Apar1from
/hal, Ike Ftynman rulesfor QED ar< unckangw.
7.48 Assuming it is he..vy enough. this 'photon' c�n deay.
(.f e�lculat� th� dec�y rate for r __ e+ + e-.
[h) If my = 300 M�VleI, find the lifetime ofthe 'photon', in seconds.
7,49 (of Find the amplitude, .A, for electron-muon scattering, in this th�ory.
(b) ealculat� the spin-averaged quantity, (1.,1111).
(el Det�nnin� th� differenti.;ol cross section for electron-muon sc�ttering in th� eM
frame. Assum� th� energy is high �nough so that the electron and muon masses
can be neglected: m,. m� ...... O. Express your answer in tenns of the incident �Iectron
energy. E and the sutt�ring angle. B.
[d) From your result in {cf, calcub.t� the total cross section. assuming the 'photon' s
i
extrem�ly heavy, myel » E,
(e) Going back to (bf. consider now the case of /o....�nergy
.
scattering from an �xtremely
heavy 'muon', Ip.I/' « .... « my « m�. Find th� difT�rential cross section in the lab
frame (muon at restf, assuming the muon does not recoil appreciably. eompar� th�
Rutherford fonnub (Exampl� 7.7), and c�lculate th� total cross section. [Actually, if
you set "i- -+ 0 and th�n take Ipl « I'I'IC, you get prt:Cisdy the Rutherford formula.)
7.5(1 (.) Find the amplitude 4, for pair annihilation (t+ + e- -+ r + rl, in this theory.
]b) Det�nnin� HAllf, assuming the en�rgy is high enough that we can ignore both the
electron and the 'photon' mass (m,. my
0).
[e) Evaluat� your result, in (b). in th� CM syst�m. Expr�s your answer in tenns ofthe
incident electron energy. E. and the scattering angle. B.
(df Find the difTerenttil cross section for pair annihilation, in the CM system, still
assuming m, = my = O. Is the total cross section finite?
7S! Spin'f particles that are electrically neutr.,] could conceivably be their own antiparticles
(if so. they are called "Major-ma" fermions - n
i the Standard Modd the only possible
candidat� art tho! nNtrinos)
(�f According to Problem 7,9. the charge conjugate spinor s
i >JI, '" irl>JI' Evidently, if
a particle is the s�me as its antip�rticl�, then >JI = t,. Show that this condition is
Lorenl2 invariant (iftrue in one inertial frame, it is true in any inertial fnmef. (Hinl:
Use Equations 7.52 and 7.53.J
(bf Show that ift = t,. the "lower" two elements of t are related to the "upper"two by
ts = -ia,oi'A' For Majorana particles, then, _ only need a two-<:omponent spinor.
X .. tAo This makes sense: A Dirac spinor takes four elements to represent the two
spin states (each) of the particl� �nd th� antiparticle, but in this case tht latter two
are redundant. Show that the Dine equation for a Majonna particle can be written
in l<omponent fonn as
. •
__
ill [ilox + i(u . V)u,X') - mcx = 0
Check that the equation you get for the "lower" elements is consistent with this.
(e) Construct spinors X representing plane wave Majorana states. (Hint: Form the
general linear combination t = al t(l) + alt(l) + �Jt(J) + a.t(·) (Equations 7.46
and 7.47f. m
i pose the constraint in part (h), �nd solve for a) and a. (in terms of a,
and all; then pkk (sayf �l = 1, al = 0 for X(I). and �l '" O. al '" 1 for X 121 .J
1213
8
Electrodynamics and Chromodynamics of Quarks
Because flu dtctromagn.etic inttractions of electrons art well undersl<!od, they scrvt:
as usejiJ probes of the structure of hadron!. Evu}'thing I said in Chap�r 7 about
leptons appJ� just as weU to quarks (using, of course, the appropriate charge: t or
- t). However, flu txptrimtntal situation is complicated by tht fact that tlK quarks
thtm.sdve5 rn:ver su the light of day, and we art obliged to inftr from the observed
behavior of mesons and baryons what their ronstiJuents art up to. In this chapUT we
shall consida" two important txampks: the production ofhadrons in tiutron-positron
collisions (Sulion 8.1), and elastic dutTon-prown scattering (Stdion 8.2). We then
turn to chromod),namics: the Feynman rults (Section 8.3J, color factors (Suticn 8.4),
pair annihilation in QeD (SWion 8.5), and asymptoticfrurkm (Sectiol1 8.6),
j
i
8.1
Hadron Production in e+e- Collisions
When electrons and positrons collide, they can (ofcourse) scatter elastically, e+ +
e- __ e+ + e- (Bhabha scattering), or they could produce two photons, e+ + e- __
y + y (pair annihilation), or - if the energy is sufficiently high - they could make
a pair of muons (or taus), e+ + e- __ p.,+ + p.,- . But they can also produce a pair
of quarks: e+ + e- __ q + q, and it is this process that I want to consider next. The
lowest-order QED diagram is
,
,
Q
,
q
For a brief moment the quarks fly apart as free particles, but when they reach a
separation distance of around lO-IS m (the diameter ofa hadron), their (strong) in­
teraction is so great that new quark-anriquark pairs are produced - this time mainly
from gluons (Figure S.l). These quarks and antiquarks (litera!ly dozens of
Inlroduction 10 EltmtnUlry Parlidts. Second Edilio". David Griffiths
CopyrightC 2008 WllEY·VCH Verlag GmbH & Co. KG�A, Weillheim
ISBN: <)78-J-S27-40601·2
276
1 8 fltClrodynamics and Chromodynomics o!QIJorlts
q
Fig. 8.1 H�dronil�tion �nd jet form�tion.
them, in a typical modern experiment) join together in myriad combinations to
make the mesons and baryons that are actually recorded in the detector - a process
known as 'hadronization'. What we observe in the laboratory, then, is �+ + �- -+
hadrons.
In all the debris there is often an unmistakable footprint leftbehind bythe original
quark-antiquark pair: the hadrons emerge in two back·to·back 'jets', one along the
direction of the primordial quark: the other along the direction of the antiquark
(Figure 8.2). Sometimes one sees a three-jet event (Figure 8.3), indicating that a
gluon carrying a substantial fraction of the total energy was emitted in conjunction
with the original qq production:
•
Fig. 8.2 A typiul two·jet event. (SOIJrct: J.
Dorfan, SLAC)
q
Fig. 8.1 A three·jet event. (SoIJ'ct: J. Dorfan,
SLAC)
• Noti<� that the quark lsay) has to 'ruel! had:· and pick up an antiqlark from the other brancb.
to make eacb jet colorien. but as lon8 ;;oS tbe energy transferred is relatively small tbis does not
disrupt the jet strucnue.
8. J Hod"," ProdUciiOll ill �+�- CO//il,'Olll
Indeed, the observation ofthree-jet events is generally regarded as our most direct
evidence for the existence of gluons.
Now, the first stage in all this (t+ + t- -+ Y ..... q + q) is ordinary QED; the
calculation is exactly the same as for t+ + e- -+ p,+ + p, - :
The amplitude is
(S.I)
where Q is the quark charge, in units of e
Exploiting Casimir's trick, we obtain
2
(lAI ) ""
(�, for u, e, and t; -� for d, s, and b).
� [(PI� r T !!,
2)
2
r[Y !'
1 + me)Y " !!'2 - me)]
xTr[y!' !!'4 - Me)y" !!,] + Me))
(S.2)
where m is the mass of the electron and M is that of the quark (Problem SJ).
Invoking the trace theorems ofChapter 7, we can reduce this to
l
(IAl ) = 8
[(PI� r
P2)
2
((PI . pJ)!P2 . P4) + (PI
P4)!P2 ' PJ )
l
l
+(me)2 (pJ . P4) + (Mc)2 (p\ . PI) + 2(me) (Mc) ]
(S.3)
or, in terms of the incident (CM) electron energy E and the angle (} between the
incoming electron and the outgoing quark:
(S.4)
(} and 1/), we obtain the total cross section (Problem S.2):
The differential scattering cross section is given by Equation 6.47; integrating over
"
=
'
rrQ
)
( )
'�
E
'
[
' - IM<'/EI' ,
1
(mel/E)2
+
"
2
( )][ ( )]
Mc'
E
'
,+
"
�'
2
E
'
(S.5)
1 277
278 1 8 f/eClrodynamicsand Cltromody/1Qmics ofQuarks
Notice the threshold at E _
Me2; for energies less than this the square root is
imaginary, reflecting the fact that the process is kinematically forbidden when there
s
i not enough energy to create the qq pair. If we are substantially above threshold
(E > Me2 » me2), Equation 8.5 simplifies considerably;'
(8.6)
As we crank up the beam energy, we encounter a succession of such thresholds first the muon and the light quarks, later (at about
the tau (at
1300 MeV) the charm quark,
1777 MeV), the bottom quark (4500 MeV), and eventual1y the top quark.
There is a beautiful way to display this structure: suppose we examine the ratio of
the rate ofhadron production to that for muon pairs:
R
==
�:i
,, ?
I ' ' '- - h "
o
"
-,
_
c
'o
�
O" ' 1
-=)
a(e+e ..... /l +/l
(8.7)
8.6 gives
RIE) � 3L:Qf
(8.8)
in which the sum is over all quark flavors with thresholds below E. Notice the 3 in
Since the numerator includes all the quark antiquark events.t Equation
-
front - it records the fact that there are three colors for each flavor. We anticipate
a 'staircase' graph for R(E), then, ascending one step at each new quark threshold,
with the height of the rise determined by the quark's charge. At low energy where
only the u, d, and s quarks contribute, we expect
(8.9)
Between the c threshold and the b threshold we should have
10 = 3.33
R = 2 + 3 ( "3' ) = '3
'
(8.10)
at the b threshold it goes up slightly,
R = 130 + 3 ( _�) 2 = 131 = 3.67
and the top quark should produce a jump to R
, This apprOximl!ion is actuaUy be!t�I
!han il looks, beaoUM of a Juclcy aJ.
gebr�i' 'lnc�il.tion: 6p,mdins th�
radical. )1 (Me!;£)2[1 + t(Mel!Elll
1 i (Mcl!E)' . . . , so the �rrOI is oford�
{Me'IEI\ nOI (Mel/Ell. As for the dutton
mass terms, th� are snu\ler 10 begin With.
though there is a second-<lrder correction;
-
=
.. 5.
(8.11)
hown�r, th�... t�rm.< cancel e""ctly in the
ca!culltion of R (Equation 8.7).
t Th� r l�plon deca)'!; predominantly into
hadrons. and this add. a bit to II. abo..., 1777
M�V; thaI'. why the exptrimenul numbers are
somewhat lbove the '.. + d + I + c· lin� in
Figu'" 8.�
8.2 Elostic Electron-ProIO" Smllering
The experimental results are shown in Figure 8.4. The agreement between theory
and experiment s
i pretty good, especially at high energy. But you may well ask why it
is not perfut. Apart from the approximation in going from Equation 8.5 to Equation
8.6 (which artificially sharpens the corners at each threshold), and the neglect of
the tau, we have made a fundamental oversimplification in asswningthat we could
treat the process as a sequence oftwo independent operations: �+ �- -jo qq (QED)
followed by qq -jo hadrons (QeD). In point of fact, the quarks produced in the
first step are not free particles, obeying the Dirac equation; rather, they are virtual
particles, on their way to a second interaction. This is particularly critical when the
energy is right for formation ofa bound state (¢ - $S, >Jt - cc, Y - bb); in the vicinity of
such a 'resonance', the interaction ofthe two quarks can scarcely be ignored. Hence
the sharp spikes in the graph, which typically occur just below each threshold.
Finally, above about SO GeV, the graph starts to rise toward the z? peak, at 91 GeV.
But, really, all this is quibbling anyway, for the importance of Figure (8.4) lies
not in what the small discrepancies whisper, but in what the overall agreement
shouts: the factor of 3 in Equation 8.8 clearly belongs there. Without it the theory
would be wildly off (look at the dashed line in Figure 8.4) - and not just at
isolated resonances, but across-the·board. That 3, remember, counts the number
ofcolors. Here, then, is compelling experilmnlal evidence for the color hypothesis
- a hypothesis that was introduced originally for esoteric theoretical reasons but is
now an indispensable ingredient in the recipe for strong interactions.
8.2
Elawc Electron-Proton Scattering
We now turn to electron-proton scattering, our best probe ofthe n
i ternal structure
ofthe proton. Ifthe proton were a simple point charge, obeying the Dirac equation,
we could just copy our analysis ofele<tron-muon scattering, with M now the mass
ofthe proton. The lowest-order Feynman diagram would be
� ')7
proton
Electron
ql
0:
�
and the (spin·averaged) amplitude would be (Equation 7.126)
(1.412) =
� L�:"lr<InLI'.
•
proron
(8.12)
1 279
2&0
I If fle"rodyl'lQmics Qnd ChromoovnQmia ofQt<orl<s
,
•
'0
•
i>-
•
u
•
•
•
,"
•
.--- ---
�
",
-"I''"
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..
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,
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t
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.
•
u
•
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!. "
•
,
-r
-..
.
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•
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= '
,
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----------
• • •
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z
. "' "3 f' ::J .E
ii t:
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.x.
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-
". -5 e� -€.
-•
..
1l ':2
=>- -£
" <=
= > j t
� 8 s:
'"
•
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• -
•
..
•
•
k�
• • . ..
... .. .. .. ..
> - <
:s � e
< "
l 'S � �
.2. C!.
cj- �
:r !;!.
_
""
g
� .. ... �
�-.;I
'0 � :; .!!.
�.. !! ';;;
.. ];
'"
1..1 0 :> C
- . .
1>
: �w
� '2:;:
0
... .. _ u
._
.'io :" !
8.2 flosl;, flwr-on-Pr-olOM SwUering
where q ::: PI - PJ and (Equation 7.128)
(8.13)
�:olOn'
only with m _ M and 1, 3 _ 2, 4). We used
(and a similar expression for
these results in Example 7.7 to derive the Mott and Rutherford scattering formulas.
But the proton is not a simple point charge. and so, long before the advent
of the quark model, a more flexible formalism was introduced for describing
electron-proton scattering. We might represent the process, n
i lowest-order QED,
by a diagram like this:
where the blob serves to remind us that we don't really know how the (virtual)
photon interacts with the proton. (However. we do assume, that the scattering is
elastic: e + P _ e + p; inelastic electron-proton scattering, e + P _ e + X, is much
more complicated, and we will not consider it in this book.) Now, the essential
point is that the electron vertex and the photon propagator are unchanged, and
therefore, since (IAtI 2 ) neatly factors (Equation 8.12),
OAti l) :::
� :�
•
L
rOJlK","
proton
(8.14)
where K"," is an unknown quantity describing the photon-proton vertex.
Well . not completely unknown, for this much we can say: it is certainly a
second-rank tensor, and the only variables that it can possibly depend on are Pl, p.,
and q. Since q - p. - P2, these three are not ndependent,
i
and we are free to use
any two ofthem; the customary choice is q and P2 (I'll drop the subscript from here
on: P "" P2 is the initial proton momentum)_ Now, there aren't many tensors that
can be constructed out of just two four-vectors; the most general possible form is
- -K1 "/<" + �
K"'" ,,,,,, q. + P · "" 1
2 p" Pv + �rJ'.
'l q. + �
6
)l \y
(MC)
(Mc) 2
(Mc
'l
(8.1S)
where the Kj are (unknown) functions ofthe only scalar variable in the problem:
q2 .* The factors (MCj-l have been pulled out, in defining K2, K., and K" just
so all the K's will have the same dimensions.t In principle, we could add an
• Noti« that p' '" (MC]l is a constant. and p! = Iq + p)l i + 2q p + '; _ {MC]l =- q p '" _ql/2.
t Th� subscript 3 is tnditionaUy rese� for a term that �nters in th� corr�sporuliMg analysis of
""..Irina-proton scatt�ring. but dots not OCCUr he..,.
=
1 281
282 1 8 Electrodynomics ond ChromodynQmics ofQuarh
antisymmetric combination
8.13),
-
b t since L"v is symmetric (Equation
(pI'q" p"q"'), u
such a term would contribute nothing to I
are not independent; it can be shown (Problem
q"K"" =
l ) . Now, these four functions
that
( Al l
8.4)
0
(8.16)
from which it follows (Problem
8.5)
that
can be expressed
of just lwo
functions,
K"V
in terms
(unknown)
K2{q2):
K"" KI (-g"" + q"'q2q" ) �
2
{Me)2 ( + �q"' (p" + �q")
Thus
=
t'
+
The form factors'
'
and
d
)
2
(8.17)
Kdq2)
and
(8.18)
tly related to the electron- proton elastic
KI K2 are irec
scattering cross section. According to Equations
and
8.13 (8.1S) (Problem S.7)
(lAlI2) = (�r {Kd(P1 . PI) 2(mc)2)+ K2 [ (PI .(��: . p) + �] }
(S.19)
in
p . (Me, 0, 0,
0). An electron with incident energy E scatters at an angle e er
with energy
(E, E') >> mel,
0);"
PI = (E/c)(l,pi)
8.8)
PI = (E'/c){I,PJ), with Pi . PI = c s and
(8.20)
-
We shall work
the laboratory frame, with the target proton at rest,
E'. Let us assume it's a moderately energetic collision
can safely g
i nore the mass of the electron (set In o (J,
(J, m
then
we find (Problem
ging
so that we
and
The outgoing electron energy, E'. is not an independent variable; it is kinematically
determined by
E and (J (Problem
8.9):
(S.21)
For a massless incident particle we have (Problem
6.10)
(S.22)
Matt formula (Equation 7.1ll) noegJ�ts proton stmeNn. and proton I"&oil; it applies to th..
but it does not aSSumr E » mil. Wr now work in thr regim� E » md. but
n.gimr
do nOi ignore proton stmcNr� and =oil (i.�. we do MOt USum� E « Mel). In the intermediate
rangr, md « E « Mol, tbe two r..:m11S agree (Problem 8.10).
• The E « Mel,
and so, for elastic electron-proton scattering
(8.23)
where E' is given by Equation S.21 This is known as the Rosenbluthformula; it was
first derived in 1950 [1[. By counting the number of electrons scattered in a given
direction, for a range of incident energies, we can determine Kl(q2 ) and K (q2)
2
experimentally. Actually, it is traditional to work instead with the 'electric' and
'magnetic' form factors, Gr(q2 ) and GM(q2);
(8.24)
GE and GM are related to the charge and magnetic moment distributions of the
proton, respectively [2].
There s
i precious little physics in all ofthis; what we have done is to set tlu agenda
for a model of the proton. A successful theory must enable us to calculate the form
factors. which at this stage are completely arbitrary. The most naive model treats
the proton as a simple point charge; n
i this case (Problem S.6)
(S.2S)
It's not a bad approximation at low energies, where the electron never gets dose
enough to 'see' inside the proton. But it is grossly n
i adequate at high energies
(Figure S.5). Evidently the proton has a rich internal strocture. That's no surprise
in light of the quark model, but it would shock anyone who still thinks the proton
is a truly elementary particle.
'.3
Feynman Rules For Chromodynamics
Quantum electrodynamics (QED) describes the interactions of charged particles;
quantum chromodynamics (QeD) describes the interactions of colored particles.
Electromagnetic interactions are mediated by photons. chromodynamic interac­
tions by g1UOI1S. The strength of the electromagnetic force is set by the coupling
constant
(S.26)
In appropriate units g. is the fundamental charge (the charge of the positron).
The strength of the chromodynamic force is set by the 'strong' coupling con­
stant
(S.27)
2St
I 8 ElU1rodynomics ondChramodynomics o/Quorks
..
.. '
..,
"
.,
, .
•
•
",
[
I'
_
1-.'.. "
' ....kI.. '
,
•
•
FiB. B.S Proton elastic form factors. Apart
from an overall constant. the electric and
magnetic form factors Gc and GM are
practiully identic.I, .nd - at lust, up to
about 10 (GeV/cf - are well fit by the
phenomenological ·dipole' function G4
(solid nne). Circles are experimental values
of GM/(1 + K}(<>:G(j. [Sou""" Fr.uenfelder,
H. and Henley. E. M. (1991) Subolomic
Physics, 2nd edn, Prentice·Hall, Englewood
Cliffs, NJ. p. 141. Based On data of Kirk,
P. N. el 01.. (1973) PIIysicol Re�ie"'. OS, 63.]
which may be thought of as the fundamental unit of color. Quarks come in three
colors,' 'red' (r), 'blue' (b), and 'green' (8). Thus the specification of a quark state
in QeD requires not only the Dirac spinor ul'l(p), giving its momentum and spin,
but also a three-element column vector c, giving its color:
(8.28)
l'U label the elements of c by a Roman subscript near the middle ofthe alphabet Ci, for example - so that i,j,Io:, . . run from 1 to 3 over quark colors.r
Typically, quark color changes at a quark-gluon vertex, and the difference is
carried offby the gluon. For example:
,
• Quarks also come in different JI<...ors, of course, but thi. i. irrdevant in QCD, except imofar as
the different quark flavor< carry different III4S$tS. Just as QED only looks at the c!u>>'ge of a p"rti·
cle, QCD cares only about its color.
t I should perhaps warn )'Ou that most book. do not specify quark color states explicitly; they are
is stage to write them
'imp��', or 'understood to be contained in w{P)'. I think it is wiser at th
out explicidy, even al the cost of some extra notational b,aggage.
8.3 Feynman Rilles For Chromodynamics
In this diagram, a red quark turned into a blue quark emitting a red-antiblue gluon.
Each gluon carries one unit of color and one unit of anticolor. It would appear,
then, that there should be nine species ofgluons - rr, rb. rg, br. bb, !?g, gr, gb, gg.
Such a nine.gluon theory is perfectly possible in principle, but it would describe a
world very different from our own. In terms of color SU(3) symmetl)' (on which,
as we shaH see, QeD is based), these nine states constitute a 'color octet':
11) = (rb + br)/...ti
12) = -i(rb - br)/...ti
= (rr - bb)/...ti
14) = (rg+gF)/./2
13)
15) = -i(rg - gr) /./2
16) = (bg + gb)/...ti
17)
= -4bg - gb)/./2
(8.29)
18) = (rF + bb - 199}/.J6
and a 'color singlet':
19) = (r:;: + bb + gg}/.J3
(8.30)
(See Section 5.5; there we were concerned with flavor, not color, but the math·
ematics is identical - just let u, d, S __ r, b, g. We're not dealing with isotopic
spin, here, and I have used different linear combinations of states within the
octet. This simplifies the notation later on.) If the singlet gluon existed, it would
be as common and conspicuous as the photon.· Confinement requires that all
naturally occurring particles be color singlets, and this 'explains' why the octet
gluons never appear as free particles.t But )9) is a color singlet, and if it exists as
a mediator it should also occur as a free particle. Moreover, it could be exchanged
i glets (a proton and a neutron, say), giving rise to a long·range
between two color sn
force with strong coupling,* whereas in fact we know that the strong force is
M�ybe the 'ninth gloon' is the photon! That
would m�ke for a beautiful unification of
the strong and d�<:tromagnetic n
i teractions.
Of course. the coupling strength isn't quite
right, but thars a problem with all unification
schemes. and could p�umably be mana�.
There's a much more serious difficulty with
this io:ka. which I"ll l� you figure out (�
Problem 8.10).
t Notice the distinction betw«n 'colorless' and
'color sin81er. Gluons 13) and 18) are color·
less, in the sense that th� n� amount of each
color is zero. but they �re not color singlets.
This situation has an analog in the themy
of spin: we can ha� a state with S, w 0, but
this does not pro"e it has spin 0 (although
spin 0 certainly implies S, .0, and by the
samt token a color singl� is necessarily col·
orless). Many authors use the word 'color·
less' to "...,,� 'color singler, but this an l�d
to misund....standing. (I was sloppy myself,
•
back in Chapters 1 and 2, b«ause at that
stage it was oot possibJ� to aplain th� idu
of a color singlet.) You might prefer the word
'color·invariant' (instead of 'color singl�') or
e\ltn 'color scalar'; the essential point is th�t
such a slale i. unaffec� by the transforma·
tions of color SU(l) (see Probl�m 8.12).
t Because gluons �re massless, they mediate a
force of infinite range (the S<lmt as tlectro·
dynamic.). In this ""nse the forc� betwttn
two quarks is actually long range. Howe�r,
confinement. and the absence of a singlet
gluon, conceab this from us. A .inglo:t stat�
(such as the proton) can only emit and absorb
a sin81� (such as the pion). so n
i di";dual glu·
ons cannot be achanged betwttn a proton
and a neutron. Thars why the force we ob·
!trw is of !I.on range. If the singlet gluon a·
i.�, it could be achanged betw�n singlets.
and the strong force �uJd ha� a compon�nt
of infinite rani/".
1 2&5
2861 8 Elurlody�amics and Chromodynamics o!QuoTh
of very short range. In our world. then, there are evidently only eight kinds of
gluons:
like the photon. gluons are massless particles of spin 1; they are represented by
a polarization vector. �j.t, which is orthogonal to the gluon momentum. p:
�j.tpj.t = 0 (Lorentz condition)
(8.31)
As before, we adopt the Coulomb gauge:t
so
�o = O.
that
�·p=O
(8.32)
This spoils manifest Lorentz covariance, butit cannot be helped (see Section 7.4). In
order to describe the color state of the giuon, we need in addition an eight-element
column vector, a:
,=
1
0
0
0
0
0
0
0
for 11).
0
0
0
0
0
0
for 17).
and so on
(S.33)
0
Elements of a will be labeled by a Greek superscript near the front ofthe alphabet
(aa): a. {3. Y
ron from 1 to 8 over gluon color states. Be<:ause the gluons them­
selves carry color (in contrast to the photon. which is electrically neutral), they couple
directly to one another. In fact. there is a three-gluon vertex and a four-gluon vertex:
• .
.
rX
Before I can state the Feynman roles for QCD. I need to introduce two pieces of
notation. First, the Gell-Mann ''\-matrices'. which are to SU(3) what the Pauli spin
• In group·thmr�ic:al t."ms. � $S
i ue he� is
and in f.cl the Coulomb gau� ""�1lO/. be
wh�her the symmetry of QCD is UP) (which
consistendy imposed. H�r. the GOrJUIK>n
would require all nine gluons) or SU(3)
to EqU.ltion 7.81 contains a factor of g,. and
(which ulls for only eight). The experimental
hence, n
i the Feynman calculus, the 'error'
situation rnolYr' the qU6tion decisi",ly n
i
fa""r of the latter.
t There is a subtle problem h."e. becauu
gau� transformations in chromodynamics
are more compUuted than EqUition 7.81.
introduced by using the Coulomb gauge can
be compensated for by appropriate modifica·
tion of the rules for compuling higher-crder
(loop) diagrams_
8.3 Feynmon Rilles For Chromodyl1Omjcs
Al = (! 001 :) (1 -00 :) (l -I00 :)
0
00 AS 00
(1 00 l) (: 0 I :0) (: !)
(
.l.
8
.
0
=
..13
(: �)
� 01 �2)
commutators A
(f"IlY)
2if"IlY.l..Y
1
512
I,
�,
8 .;3/2
matrices are to 5 U(Z):
,
=
=
,'
.1..4 ""
,'
0
.1..(, =
=
(8.34)
'
).,7
=
1
the group 5 UP):
Second, the
(A"', AIl)
ofthe
matrices define the 'structure constants'
=
(summation over y - from
of
(8.35)
to 8 - implied by the repeated index). The structure
constants are completely antisymmetric,fllay _farll _ _fallr. You can work them
out for yourself (Problem 8.15). Since each index runs from
x
8
_
I to 8, there are 8 x 8
structure constants in all, but most of them are zero. and the rest can be
obtained by antisymmetry from the following set:
pu =
rS
f'78
=
p47
=
f2<6 = pS7 = f)05 = fl�
= f'17 =
=
c:
(8.36)
I can now state the Feynman rules for evaluating tree-level diagrams in QeD:
1. Exlu7I111 Lin�. For an external quark with momentum p, spin
s, and color
Quark :
j
Incoming {--.) :
ul'l(p)c 1
(8.37)
Outgoing ( ....)... . uI')(p)ct
(note that ct = c' will be a row matrix). For an external
anliquark:
Antiquark :
j
Incoming ( __) : v!>I(p)Ct
Outgoing ( ....)... : v('l (p)c
1
(8.38)
where c represents the color of the corresponding quark. For
an external gluon of momentum p, polarization 1:, and color
1281
288
1 8 EI�ctrodyl'OQmiG' "ndChromOOynamics afQuarks
a,
include a factor
(To avoid confusion it is helpful to indicate on the diagram
the indices - space·time and color - you are using for each
gluon.)
i temal line contributes a factor
2. Propagators. Each n
(8.4Q)
,
"""""
CIe, p.
i3, •
) ..
_.
/j"'/l
,""
<f
(*) -�8<
(8.4l)
3, Vertices. Each vertex introduces a factor
Quark.gluon:
:
' :}
�
Thra.gluolI: ( ., ,
., �
Y. .
).'" y"
(8.42)
:
_gJ"/lY[&...(kl - k2)A + go,,(k2 - k))" + 8>.,, (/,:) - k1).]
(8.43)
Here the gluon momenta (/,:1. k2, kJ) are assumed to point
into the vertex; if any point outward n
i your diagram, change
their signs.
�') :
..�.
FourgIUQII: ( '
-ig? fJ"/l�fyS� f.&,;.g.P - gl'Pg.,,) +rS�fJJY�(g.,.g;..., - g,,;.g.p)
(8.44)
(summation over T/ implied) .
Everything else is the same as for QED'; impose conservation of energy and mo·
mentum at each vertex to determine the internal four momenta; follow each fermion
line 'backward' along the arrow, erase the overall delta function, and multiply by i to
get A. In the nexttwosections I'll workout some examples to show you how it goes.
• Loop di.agums in QeD require special rules. including the introduction of so<alled
'Fadd�v-Popov ghost.'. The� art d�p water•. n
i to which _ shall not venture [3J.
8. 4 Color "octors
'A
Color Factors
In this section, we consider the interaction between two quarks (also a quark and
an antiquark) in lowest-order QCD. Of course, we cannot observe quark-quark
scattering directly in the laboratory (although hadron-hadron scattering is an
indirect manifestation), so we won't be looking for cross sections here. Instead,
we concentrate on the effective po�ntials between quarks - the QCD analog of the
Coulomb potential in electrodynamics. We used such potentials, with a promise to
derive them later, back in Chapter 5, in the analysis of quarkonium. Bear in mind
that this is a perturbation theory calculation, valid only insofar as the coupling a,
is smalL We cannot hope to get the confining term in the potential by this route
- we are implicitly relying on asymptotic freedom, and all we're going to find
s
i the short-range behavior. Nevertheless, we wi
l l obtain a very suggestive result:
Quarks attract one another most strongly when they are in the colorsingltt configuration
(indeed, in other arrangements they generally repel). At very short range, then, the
color singlet is the 'maximally attractive channel' - an indication that binding is
more likely, at least, for singlet states.'
8.4.1
Quark and Antiquark
Consider first the interaction of a quark and an antiquark. in QCD. We shall
assume that they have different flavors, so the only diagram (in lowest order) s
i the
one in Figure (S.6),t representing, for instance, U + d ...... u + d. The amplitude is
given by
(S.4S)
{'A6)
(summation over ct implied). This is exactly what we had for electron-positron
scattering (Equation 7.108), except that g. is replaced by g, (ofcourse), and we have
singlet, Or that it CDnnol occur in other configurations. For this we would have to know the
Iong·range beh..vior of the po�nlial, aboUI which, a� present, we can only Sl""ula�e.
t In principle, for the S4"'" fbvor (e.g. u + ;; -> u + U) we should include a second di..gram, as
in e\ectron-JIO$ilrOn scattering (Figure 7.5). How�. in the nonrelativistic �mit of inte�st here
this second diagr.om does I"IOt contribute anyw�y (see foo�no�e to Example 7.3), so in practice
wha� we're doing applies just as wdl whala'U the quark flavors. (See also Problem 8.17)
• This is � very ple.sing conclusion, bu� it d�. I"IO� pro,"" th�t binding musl occur in the color
1 289
00
I 8 EiedmdynomicsondChromodynomics ofQuorh
Fig. a.6 The quarlc-antiquarlc int�raction.
in addition the 'color factor'
(S.47)
Therefore, the pou:ntial describing the qq interaction is the same as that acting
in electrodynamics between two opposite charges (to wit: the Coulomb potential),
only with a replaced byfa,:
"''''
Vqq(r) = -f,
(8.48)
Now, the color factor depends on the color state of the interacting quarks. From
a quark and an antiquark we can make a color singlet, Equation 8.30, and a color
octet, Equation 8.29 (all members ofwhich yield the samef). I'll calculate the octet
color factor first, because it's a little easier [4].
Exomple 8.! Color Fo�orfor the Octet Configurotion A typical octet state (Equation
S.29) is rb (any of the others would do just as well; see Problem 8.16). Here
the incoming quark is red, and the incoming antiquark is antiblue. Because color
is conserved: the outgoing quark must also be red and the antiquark antiblue.
Thu,
and hence
•
y�s. quuk color can chang� at a QeD vertex, but in this ca.o;o, the outgoing antiquark cannot
carry ofT the positive unit of redness. so the outgoing quark is forced to do that job.
8.4 Color Foctors
A glance at the .\ matrices reveals that the only ones with entries in the 1 1 and 22
positions are .\1 and .\8 . So
Exampl� 8.2 Color Factor for the Singlet Configuration The color singlet state
(Equation 8.30) is
(1/../3)(r' + bb + g8)
[f the incoming quarks are in the singlet state (as they would be for a meson, say),
the color factor is a sum of three terms:
The outgoing quarks are necessarily also in the singlet state, and we get ninc terms
in all, which can be written compactly as follows:
I I I ,, �
f = - - - (.l. .. .\.. )
/
L
,
/3 � J'
4 ..
1
.\ )
= -Tr(.\
<It
"
12
(8.50)
(summation over j andj, from 1 to 3, implied in the second expression). Now
(8.51)
(Problem 8.23), so, with the summation over a (from 1 to 8),
Trp,�.\") = 16
(8.52)
Evidently, then, for the color singlet
f= �
(8.53)
3
..
Putting Equations 8.49 and 8.53 into Equation 8.48, we conclude that the
quark-antiquark potentials are
4 ex,tic
VoN(r) = - - 3
VoN!r)
,
(colorsm
. gJetl
(color octet)
= -6 ,
1 ex,tic
(8.54)
(8.55)
1 291
292 1
8 fkClrod)inQmiCl Qnd C�'Qmod)i"QmiCl QfQ�Q,/<j
Fig. $.7 The qu�rk-quark interaction.
i the color singlet but rt:pulsivt
From the signs it appears that the force is atlractivt: n
for the octet. This helps to explain why quark-antiquark binding (to form mesons)
occurs in the singlet configuration but not in the (color) octet (which would have
produced colored mesons).
8.4.2
Quark and Quark
We tum now to the interaction of two quarks. Again, we shall assume that they
have different flavors, so the only diagram (in lowest order) is the one indicated in
Figure (8.7),' representing, say, u + d --., u + d. The amplitude is
g
- � I
..II "" -t ;p[ii(3)yl' u(I)][ii(4)yl' u(2)](cI)..
"cl)(c1).."C2)
g"
(8.56)
This is the same as for electron-muon scattering (Equation 7.1(6), except that
is replaced by
f=
and there is a color factor
�(cI)""cd(c!}'''C�)
g.
(8.57)
The potential, therefore. takes the same form as that for like charges in electrody·
namics:
.,,"
V'I'I(r) =/-
(8.58)
,
Again, the color factor depends on the configuration of the quarks. However, from
two quarks you can't make a singlet and an octet (as for qq) - rather, we obtain a
triplet (the antisymmetric combinationS):
I (bgI'" -bgb)'l/fij./2 I
-
IE' - '1<i1fi
(triplet)
(8.59)
i clusion of this diagram. to­
• For id�ntical <JuarKs th�r� is also a 'cros�' diagram. H()W�""r. n
gether with the statistic.ol factor S in the cross s«tion formula. leads to the same non�b.tivi•.
tic limit (Stt footnote to wmple 7.3). so n
i fact our potentials are COrr«t even for same-flavor
quarks.
8.4
Colar FOClon
and a sextet (the symmetric combinations);'
I
(rb + br)/.J2, (bg + gb)/.J2, (gr + rg)/.J2
rr,bb,gg,
I
(8.60)
(sextet)
Exampl� 8.3 Color FaClorfor the Senel Configuraljon A typical sextet state is rr (use
any of the others ifyou prefer - you'll get the same result for1). In this case
( )]
I
1
o Op.. � = 4"P.uA!l)
Jj Jj 1
l l + A11A11
8 8 )= I [
= 4"I (A11Al1
4" (1)(1) + (1/ 3) (1/ 3)] = "3
O!
Exampl� 8.4 Color Factor for th� Triplet Configuralion
(rb - br)/.J2, sot
(1
0
O)'"
IO
1
0jA"
11
0
0)'"
10
1
0 A"
II
1
O!
(8.61)
..,
A typical triplet state is
m] [ (:) ]
+ (:)][ 0jA" m]
+ m ][ m]
+ j (!)] [ m]1
f � l ;, ;,
l[
O!
0)'"
0
0)'"
0
0)'"
• III graup-thtordical langwlge, 3 ®3 = I ffi8. but 1 ® 3 = JIB 6.
t H�.., (rb _ br) .... (rb - br), so thert ar� fOUT tefIIls. S'h�m.ati,any. rb ..... rh. rb ..... -br. -br .....
rb. and -br ..... -br (in Ih� lasl lerm the f"ClOT< or -\ cancel).
1 293
2941
8 EI,arodynamic< Qnd C"romodymlmi� afQuarln
�P'I\),i2 - ),12),iJl
= � � .l..1l + ),�l),�l - )'ll.l..ll ),�l),�l)
= � -1 + � - 1 - 1) = - �
=
-
(), 1
(
(8.62)
...
Putting Equations 8.61 and 8,62 into Equation 8.58, we conclude that the
quark-quark potentials are
V'i'!(r) = - 3 ,
2 a,fu:
(color triplet)
1 a,fu:
V'I'I(r) = 3 -,
(coIor sextet)
(8.63)
(8.64)
[n particular, the signs indicate that the force is attractive for the triplet and reo
pulsive for the sextet. Of course, that's not too helpful as it stands, beause �ither
combination occurs in nature: However, it does have interesting implications for
the binding of three quarks. This time we can make a singlet (completely antisym·
metric), a decuplet (completely symmetric), and two octets (of mixed symmetry),
as we found in Section S.6.1.t Since the singlet s
i completely antisymmetric, every
pair of quarks is in the (antisymmelric) triplet state - the attractive channeL In the
decuplet, every pair is in the (symmetric) sextet state - they repel. As for the two
octets, some pairs are triplet and some are sextet; we expect some attraction, then,
and some repulsion. Only in the singkt CQnfigumtion, though, do we get complete
mutual attraction of the three quarks. Again, this is a comforting result: as in the
case of mesons, the potential is most favorable for binding when the quarks are in
the color singlet configuration.
••S
Pair Annihilation in QeD
In this section we consider the process quark plus antiquark ...... two gluons - the
QeD analog of pair annihilation. The calculation is quite similar to Example 7.8;
• If�ou don't heed the mrning in footnote
(*} to the fim paragraph of S«lion 8.4, you
may b.. abrffiM to find that two quarks in
the triplet state attract one another. There is
some comfort n
i the obiOet\'alion that tlw: sin·
glet qij coupling is Iwiu as ,frong: but slill, if
this were the whole story we might very well
expect tripld '1'1 binding to occur. leading to
free 'diquark' states. There has. in fact. been
some speculation about the possible existence
of diquarks within nuclei (S).
t In Chapter S we were dealing withjlaVOT. not
,%t. but the mMht"",liu is the same. Group
theoretic:olly. 3 0 3 0 3 R I (fl 8 $ 8 (fl lO.
8.5 Pair Annihilation in QeD
however, in QeD there are thru contributing diagrams, in lowest order:
(2)
OJ
(3)
(I)
The amplitude for the first diagram is
�I
(2)
(3)
,
;/
(')
" ,
y, ,
q
0)
j� ,
�
'. ,
(3)
[ + 1m) 1
i f
Il. 'l 'f· fl', (f
= iii{',,'2 (- i�,l.
2 Y 4" . q2 m2c1
x
[-i�,l.". yl'] [fjl'arlu(1)cl
(S.65)
(fo simplify the already overburdened notation I'll leave the * off the gluon
polarization vectors and color states until the end.) Here q - PI - Pl , so
(S.66)
and hence
Similarly, for the second diagram:
_g,1
.41 = - -- {ii(l)!ll(fl -"14
S
1
PI ' p.
+
II I
me)/.]u(I)) aja4(Ci,l..
0",l..ll eL)
(8.68)
Notice that the A's appear this time in the opposite order. Finally, for the third
diagram:
At} = iii(2)ci
[-i�A�Ya] u(l)el [_it'�&Y ] (_gJ"""Y[&'.(-PJ + p.h
+g"�(-P. - q}l' + g..,, (q + Pl)"lHfra)"][l:�]
In this case q - Pl + p., so tT = 2Pl . p.; simplifying (and using
we find (Problem S,20):
At)
(8.69)
f) p) = f. P4 = 0),
.
'
v(l)[(fJ . f.)(f. -"11) + l(Pl . f.)/J - 2(p. ' fl)I.)u(l)
= i-4 -p} . p.
.:
1
xr"r,,)",,!(c1Arcd
(S.70)
1295
296
I 8 Eiearodynomin ond Chromodynomies ofQ!Jorlu
So far, this is all completely general (and rather messy) . To make things more
manageable, let's assume (as we did in our study of e+ e- annihilation) that the
initial particles are at rest:
PI = P2 = (me,O),
PJ "" (me,p),
(8.71)
P. = (me, -pI
(8.72)
Meanwhile, in the Coulomb gauge (Equation
8.32)
PJ · f4 = -p · t_ = -p4 ·f4 = 0
(8.73)
(likewise P4 fJ = 0), sotwotenns n
i AJ drop out. Using Equations 7.140 and 7.141
.
to simplify Al and A1, we find that the total amplitude (A = Al + A1 + Al)
can be written
,
A = - � a)a:ii(2)cI lfd4"_AaAti + 14 1l IlAtlAa
8(mel
-i(fl · E'4) jf_ -I,)j'tlYAYJelU(l)
We may as weI! orient our coordinates so that the z axis lies along p; then
(8.74)
(8.75)
From Equations
7.145 and 7.146 we have
Putting this into Equation 8.74, and exploiting the commutation relation (Equation
8.35) for the A'S, we obtain
,
A = �aJa!ii(2)Cr[(tl . f.){Aa , AtI }yO
8m,
+i(fl x f.j . l:(W, AtlJyO + {Aa, AtI}y }]el
l
utI}
(8.77)
where curly brackets denote the anticommutator: (A, B) '" AB + BA You might
compare this result with the corresponding expression in QED (Equation
to which it reduces f
i you set aU the A'S equal to 1, drop the color states a and c, and
letg,/2 -+ g
Suppose now we put the quarks into a spin-O (singlet) state (the triplet state
cannot go to two gluons anyway; it needs at least
7.146),
•.
three);
(8.78)
For A, J. we have (Equations 7.153 and 7.154)
(8.79)
As before, AH "" -An, and we are left with'
A
""
,
-i.J2� (f:J x f()'aJ"a�(cj(,\", ,\Illed (spin singlet)
(8.80)
Once again, we have obtained a result that is identical to the one in QED (Equation
7.158), except that g, _ g and there is a color factor
"
(8.81)
In particular, f
i the quarks occupy the color singlet state, (l/.J3)(rr + bb + 88). then
f
�
�,;,� ;, 1(1 0 0)1>- '"
+100 111'-" "
But
m
(:) I ':"
�
8
+ 10 1 011'- '"
';<!.T'I'-" "
(Problem 8.13), so
(color singlet)
m
(8.82)
(8.83)
Now, the singlet state for two gluons (see Problem 8.22) s
i
(singlet) = .....rs L In)dn12
1
•
-,
(8.84)
"
Evidently
(8.85)
, At this St:l� aU t." ms in f, f. drop out. The fact that At, is proportional !Q 'J ., (Equation
8.74) meanS that the diagram containing a thI_gluon verta makes no contribution. when the
quarks are at rest in the spin singlet configuration. Most books 'imply ignore it from the start,
but in principle il should he included (see Problffil 8.21).
298 1 8 Elutror/y"omics u"dChromor/yllum;os ofQ!Jorlc,
and hence
(8.86)
i the spin singlet, color singlet configuration,
Conclusi<m; For q + ij -+ g + g n
with the quarks at rest, the amplitude is
.A
= -4j2jj�
(8.87)
(compare Equation 7.163). and the cross s«tion is
(1
=
� 41r
3 cv
(,a,)'
(8.88)
m
(see Equation 7.168). Just as the cross section for e+ + e- -+ y + y determines the
positronium de<:ay rate
thru
(8,89)
(Equation 7.171), sowe can now givea formula for thedecayofa spin·Oquarkonium
state, such as 'Ie (note that 1fr and Y themselves carry spin 1, and go to
gluons):
(8.90)
As it stands, this is not terribly useful, since we don't know 1fr(O}. However, the
electromagnetic decay '1< -+ 2y involves the same factor, and we can derive a clean
expression for the branchn
i g ratio (see problem 8.23) .
•••
Asymptotic Freedom
In the last section of Chapter 7 we found that the loop diagram
8.6 Asymptotic Frudom 1 299
in QED makes the effective charge of the electron a function of the momentum
transfer q:'
The coupling strength increases as the charges get closer together (larger Iq�II, a fact
that we interpret physically as a consequence of'vacuum polariZiltion': the vacuum
functions as a kind of dielectric medium, partially screening the charge. The closer
we approach, the less complete is the screening, and the greater is the effective
charge. of course, Equation 8.91 is valid only to order a (0)2. There are higher·order
corrections, ofwhich the dominant ones come from chains ofbubbles:
q
,
,
• • •
q
As it happens, these can be summed explicitly, and the result ist
2
a(lq
II = CI---'
[.-' M
"
.
'7
In
"
I /J
lIq
(O-'
"
I'(
'''
I/''
("
(�
=
a(O)
(8.92)
Ostensibly, the coupling blows up at In!lq21/{mc)2 ] _ 3.1l"ja(0). However, this is not
to be taken too seriously, since it occurs at an energy ofabout 10280 MeV, which (to
put it mildly) is not an accessible region (see Problem 8.24).
• It also introduces a div<rgtm te<m. which
we so,aK up in th� 'r�nornuliud' char�
(Equation 7.189). But Ihars an �ntir�ly differ·
�nt probkm, on� thaI (how��r lroublesom�
)'<Iu may find it in prindpl�) has no obs�rv·
abl� consequ�nces. and OnCe the appropriate
n
i cantation has �n mad�, is of no further
significance. The perfectly fin;� dependence
of 0 on r/ is Ih� rigrlijico.m matter. for it
carries direcl and m�asur:l.bl� implications.
t This is not 10 surprising. What we have, in
�lfecl, is th� gwmetric series
1 + x + x2 + xJ + . .. = __
I -x
where x is for one bubble. ,,' is for two. and
so On. Although Equation 8.92 is correcl to all
orders in 0(0). it is nat txaet. since we are ig·
noring diagram. such as
q
q
These can be shown to make a much smaller
contribution in the limit Ir/I » (me)l.
Equation 8.92 is known as the 'leading log'
approximation.
300
I 8 EI�>1rodyI1Qmi<;! and Chromodyl1amics o/Quark!
Much the same thing happens in QCD: quark-antiquark bubbles
lead to a screening of the quark color, which (modulo appropriate color factors) is
the same as Equation 8.91. However. there is a new twist to the story, for in QCD
we also have virtual gluon bubbles
as well as diagrams of the form
It turns out [6] that the gluon contribution works in the other direction, producing
'antiscreening· or 'camouflage'. I do notknowofa persuasivequalilative explanation
ofthis effect [7] - suffice it to saythat the formula for the running coupling constant
in QeD (analogous to Equation 8.92) is
(8.93)
8.6 Alymploli<; Freedom
where II is the number ofcolors (3, in the Standard Model), andf is the number of
fl.avors (6, in the Standard Model). In any theory for which 1111 > 2f. antiscreening
will dominate, and the coupling constant will decrease with n
i creasing Iii; at short
distances the 'strong' force becomes relativelyweak. This is the source of asymptotic
freedom. on which so much of what we can say quantitatively about the hadrons s
i
predicated. Asymptotic freedom is what licenses the use of the Feynman calculus
in QCD to calculate interquark potentials; it is a basic ingredient in the theory of
quarkonium; and it is presumably responsible for the OZI rule. Chromodynamics
would have gone out of business if it had not been for the timely discovery of
asymptotic freedom {8].
You may have noticed the appearance of a new parameter, p., in Equation 8.93
In ele<:trodynamics it s
i natural to define 'the charge' ofa particle as the long-range
(fully screened) value - that's what Coulomb and Millikan measured, and it's what
an engineer or a chemist or even an atomic physicist (unless he's measuring the
Lamb shift) is concerned with. Thus a(O) is the 'good old' fine structure constant,
1/137, and it is the sensible parameter in terms of which to do perturbation
expansions. But we don't have to do it this way; we could work from any other
2
value of q (provided only that we stay well below the singularity in Equation 8.92,
where a(lil) runs larger than 1, and perturbation theory breaks down). In QCD,
2
however, we alllno! work from q
_
0, because that's where a, is large. We must
use as a reference some place where as is small enough to justify a perturbation
expansion. That's why Equation 8.93 is expressed in tenns of O'.(p.2). instead of
0',(0) . Provided that it's large enough so that O',(p.2 ) < 1, it doesn't matter what
value of Jl. you use (see Problem 8.25). Indeed. if we introduce a new variable A,
defined by
(8.94)
the running coupling constant can be expressed in terms of a sillg!e parameter:
(8.95)
(see Problem 8.26). This compact result tells us explicitly the value of the strong
i hard to
coupling at any Ii i. in terms of the constant A. Unfortunately, it s
determine A precisely from experimental data, but Ac appears to lie somewhere
in the range
100 MeV < Ac < 500 MeV.
(8.%)
Notice that whereas the QED coupling varies only minutely over the accessi·
ble energy range (Problem 8.24). variation in the QCD coupling s
i substantial
(Problem 8.27).
I lOt
3021 8 Elutrodynamics ami Olromodynomics 0fQuorks
References
1 Rosenbluth, M. N. (19S0) Physi.:al JU.
view, 79. MS.
tricks. See Perkins. D. H. (1982) In·
troducti<on 10 High·Entrgy Physics,
2 See. for elIample. Frauenfelder. H.
and Henley. E. M. (1991) SubalOmic
Physics, 2nd edn. Prentice·Hall, En·
glewood Cliffs, N.J. Ch.1pter 6. I
should warn y<>u that this subjed is
a notational nightmare in the liter·
ature. There are only two variables
2nd edn. Addison·Wesley. Read·
ing, M.A. Appendix G (in the 3rd
edition (1987) it is AppendiJc J. and
in the 4th (Cambridge, UK: Cam·
bridge University Press, 20(0) it
h.1s been removed); (a) Halzen. F.
and Martin. A. D. (198-4) Quari:s
and I.eplOns, John Wiley & Sons,
New York. p. 211. Section 2.15; (b)
Quigg, C. (1997) Gaugo Throm. of
I"" Strong. Weak. and Electramag-·
ndic Inleractions, Addison Wesley,
Reading, M.A., pp. 198-199. I pre·
n
i the problem - the incident elec·
angle (0) - but it is common to en·
troo energy (E) and the scattering
of E, £'.I:I.i(. QI .. -i(. r ..
-i(/4Mlel '" P ·f/Me, w . -2p ·
q/i(, W ,. (q+p) , x _ -i(/2p .
q. and y ... p q/p Pl. Moreover.
counter a more or less random mix.
ture
J
fer to do it by the more pedestrian
although there are only two in·
method shown here, which is close
dependent form factors involved,
in spirit to the approach of; (c) Kane.
G. L. (1971) Color Symmetry and
QI«1ri: Confinement, Proceedings ofl""
Ill" R.,.con/re de MorWnd. voL III.
there are many different ways to ex·
press them. Some authors favor F,
and FI, with K, • -i({F, + KF2)l
and K2 "" (2M,)l Ft - K2qlF� (K .
ed. J. Tran Thanh Van. Frontieres,
1.�28 is the 'anomalous' contribu·
tion to the proton's magnetic mo­
ment): others prefer G f . F, - Kr
Fl. GM F, + KF2. (The latter are
related to the Fourier transforms
•
of the charge and magnetic mo­
ment distributions. respectively; see;
(a) Halzen. F. and Martin. A. D.
(In.) Quarks and I.eplOns, John Wi·
ley & Soos, New York. Sect. 8.2.)
Anyone can play this game;
K"
and Kl are my own contributions.
3 The interested reader should
consult the classic treatise by
Abers, E. S. and Lee, B. W.
4 People in the know seem to be able
(1973) Physics Report" 9 C, 1.
to c.alculate color factors on their lin·
gers; everyone h.1s his or her own
Gif·sur·Yvette. France. p. 9.
� dose, F. {1982) " Demon Nuclei·',
Nmurr, 296, lOS.
6 Quigg, C. (1997) Go."8l' Throm. of
I"" Slrong. Weak, o.nd E/a;tromo.gnelic
Inl.,,,,lio,,,. Addison Wesley. Read·
ing. M.A., pp. 198-199. Sect. 8.3.
7 See. however, Quigg. C. Gauge Tho·
aries ofthe Strong. Woo.k. o.nd Elutro·
ma.gn.etic Inl<:=lio"" Adds
i on Wes·
ley. Reading, M.A., p. 223; and (April
1985) Scientific American, p. 84.
S Politzer. H. D. (1973) Physical
R",,1I;w I.elteN, 30. 1J.4<i; (1974)
Physic, Roports, 14C, no; (a)
Gross D. J. and Wilczek. F. {1973}
P"ysiad Review Letters, 30, 1343.
Problems
..,
(IJ Derive Equation 8.1, from the Feynman rules for QED.
(h) Obtain Equation 8.2 from Equation 8.1
(cJ Derive Equation 8.1 from Equation 8.2
(dlDerive Equation 8.4 from Equation 8.3
8.2 Derive Equation 8.5, starting with Equation 8A
S.l Why don't we use (T (o+e- ..... e+e-) in the denominato ,
r to define R (Equation 8.7)?
8.6 Asymptolic Freedom
U Prove Equation 8.16 [Hin/: First show u...t q.. O"=O. Then argue that we
may as wdl
take K'" such that q" K"'=O, in the sense that any term in K'" u...t does "01 obey
q.. K"',.,O will contribute nothing to L'" K....] Comme.." Equation 8.16 actually follows
more simply and generally from charge conservation at the proton �rtex. but I ha�
not de�loped the formalism here to make this argument (see Halun and Martin (21,
Sections 8.2 and 8.3).
(One way to proceed is as follows. Take If' = (0. O. O. q); then q"L'" = 0 => L'" =
UD
So L"'�.
�
be zero.]
un PD
,od ili"
, m;gh< " �Il
8.5 Prove Equation 8.17. from Equation 8.16 [Hi..,: First contract K'" with q.. , then with
p ]
U Find KL and K2. and also GE and GM. for a 'Dirac' proton (Equation 8.2S).
8.? Derive Equation 8.19
U Deri� Equation 8.20
U Derive Equation 8.21
S.lO Ched: that the Rosenbluth formula (Equation 8.23) agrees with the Mott formula
(Equation 7.131) n
i the intermedi;ote-energy regime (mel « £ « Mol). Use the
expressions for Kl and Kl appropriate to a 'Dirac' proton (Problem 8.6).
s
w
u: The gluon would couple to all
S.H Why can't the 'ninth gluon' be the photon? [A..
baryons with the Sime strength, not (n the photon does) in proportion to their charge.
Since mass and baryon number are approximately proportional in bulk matter, such a
force would, in fact, look very much like an extra contribution to gravity. There was a
Rurry of interest in this possibility in early 1986. (fischbach, E. d aL, (1986) PhyW.<ll
Rwiew Wltrs, 56, 3. See, however, the comments in /'kysical Review !.elltrs, (1986) 56,
2423.]
8.12 Color SU(3) transforrmotions rebbel 'red', 'blue', and 'green' according to the transfor­
mation rule
•.
c -+ r! = Uc
where U is any unitary (Uut =1) 3 x 3 malTix ofdeterminant 1, and , is a three-element
column vector. For �mple
would take r -> g, g ..... b, b ..... r. The ninth gluon (19)) is obviously invariant under U,
bUI the octet gluons are /tOt. Show u...t 13) and 18) go into lineu combinations of one
another:
13') = 0:13)
+
.BIS},
IS'} = Yl3} + JIS)
Find the numbers ..., fJ, y, and 8.
8.U Show that
(Notice that all the ).. rmotrices are tracelells,)
1)03
31M
I 8 Elearodyrwmics andC"romodyn�mics oJQuarks
3.14 What ar� th� smlCtur� constants for SU(l)? nut is, wn.t ar� ttu, numbersJ;;' in
5.15 (a) Giv�n tn.t r�r is compl�U'ly antisymm�tric (so that PII=O automatically, and
having calculatNJlll, w� don't need to both�r with.fll.f2ll, �tc.) how many disll...:!
nontrivial structure constants r�main?
[
8·'·' 1
Answer : --- = 56
3·2·1
Of theR, it turns out that only nin� are nonzero (those listed in Equation 8.36), and
among tMs� th�� ar� only thr� diffe�nt "umbers.
(b) Work out (I..L , J.l), and confirm thatplr=o for all y except 3, whil�pll=l.
lc) Similarly, compute (J.I• J.l) and IJ.4. J.�), and d�tennine the resulting structure
constants.
1.16 Cakulat� th� octet qq color bctor using the state
(.) bg
(bJ(r' - bb}/Ji.
(c) (r' + bb - 2gg)/../6
1.17 Find th� amplitude.K for the diagram
What s
i th� color factor (analogous to Equation 8.47) in this u�? EvaluateJ in th� color
singlet configuration. Can you explain this result? IAnswrr: It's ura; a singlet unnot
coupl� to an octet (g1uon).1
5.18 Calculat� the sextet qq color factor using the stat� (rb + br)/Ji..
I.U Color factors always involve expressions ofthe form J.;J..� (summN over "'). There is a
simple formula for this quantity. which shortens the uithmetic:
(a) ;=j=l=I=l (see Equation 8.61)
(s� Kan� (4)). Ched: this theorem for
(b) ;=J".l, k�I..2 (s� Equation 8.49)
(e) ;=I=l.j",k",2 (s� Equation 8.62)
'"'
(<I) Use it to confirm Equation 8.52
5.20 Derive Equation 8.70, starting from Equation 8.69
HI There is a simple test for the gauge n
i variance of an amplitude (.4') in QeD (or
QED): Replace any g1uon (or photon) p;>Iarization vector by its mom�ntum (€J .... PJ,
say). and you must get zero (see Probl�m 7.23). Show using this criterion that .K ,..
..Ifl + ..If) + ..If) is gauge·invariant, but.At'1 + ..Ifl alone is not. [Thus the thr�·g!uon
verte� is essential in QCD to pr.-s�!Ve gaug� invariance. Notice, by contrast, that
..Ifl +.ltl alone is gauge-invariant in QED (ElLample 7.8)_ The fact that J. matric.-s do
not commute mak.-s tht diff�rence.)
&6 kym�lic Freedom
8.22 Construct the color singlet combinatinn oftwo gluons (Equ�tion 8.84). One method is
as follows: Let
Under SU(3), G ..... c'""Uc. where U is a unit.;ory matro: of determinant 1. Similarly, let
dI"=(i', b, g), transfo ming by the rule dI" ..... rft=dI" ut. Form the matrix
r
N � M - jrrr (M)], so that Tr(N) = 0
I
[fr(M')=Tr(M)=(rr + bb + g8). so this combination s
i SU(3)·invariant: it is the singlel
combination in 3 @ 3" 1 GIS. N is the octet.] Note that
N' = M'
=
- �(Tr(M')] :: UMUt - �rrr(M))uut = UNUt
nus tells us how the giuons them�lv� (which are intheoctet repr�ntatioo) transform
under color SU(3). The question is how to put together Iwo octel$ to make a singlet:
that is. how to make somethins bilinear in Nt and N2 which is invariant under U."I"IW'
solution s
i
It remains to figure out what 5 is in tenns of the elements of M) and M2:
Tr(NJN1I = Tr( (MJ - "3 [Tr(M1)])(M1 - jrrr(Ml)])]
I
I
I
= Tr(M1Ml) - "3(Tr(Md]fTr(M1)1
= j [(rrh (rr h + (bb)t(bb)z + (gg}t(Uhl
-
2
1
-
-
-
-
- j [(rr]t{bbh + (rrh(ggh + (bbh (rrh + (bbh(gg)z
+(ggh(rrh + (ggh(bbhl + [(rb)J(br)l + (rgh(grh
+(br] l (rbh + (bg)l(gbh + (gr]l(r;gh + (gbh(bghl
1 305
3061 8 EiecuodyMmics lmd Chromodynumi<3 ofQuorlc.
= Ilhllh + 12)112h + 13)d3h + 14)d4h
+IShISh + 16hl6h + 17)d7h + 181J18)z =
•
L In)dnh
�=1
(To normaliz� � state, divide by ./8.) lllis - w invui�nt product oftwo octtts - is
the SU(3) �rnolog to the dot product oftwo 3·v«tors in SU(2).
8.23 Dttermine the branching utio r (�< ...... 2g)fr (11< ...... 2)'). IHint, Use Equation 8.90
for the numer�tor, and a suitable modific.ation of Equations 7.168 �nd 7.171 for the
denomirnotor. There are two modifiutions: (i) w qUilrk charge is QI: and (ii) there is a
color f�ctor of3, for quarks in the singkt state (Equation 8.30). A1>SIVo:r :
I{a,/ai.)
8.24 'a) Calculate the energy (Jjq!jCI) at which the QED coupling constlnt (Equation 8.86)
blows up. (Remember, a(O)=lf137, th�fin�structure constant.)
(b) At what energy do we get a 1% dep.;orture from a(O)? Is this an accessibl� �nergy?
8.25 Pro� that the value ofI' in Equation 9.69 is arbitrary. [That is, suppose physicist A uses
th� valu� 1'., �nd physicist B uses a diff�r�nt value, Il• . A...,,_ A's �rsion of Equation
9.69 is corred, and prove that B·s s
i also correct.]
8.26 Dtrive Equation 9.71 from Equations 8.93 and 8.94
8.27 Calculate ", at 10 and 100 GeV. Assume A=0.3 GeV. \Vbat if Ac=l GeV- How about
Ac=O.l GeV?
8.28 (Gluon-g[uon scattering)
[a) Draw the )owest-order diagrams (�r� are fOUT ofthem) representing W interaction
oftwo gluons.
[b) Write down the corresponding amp[irudes.
(e) Put the incoming gluons into the color singlet state: do the same for the outgoing
g!uons. Compute the resulting amplitudes.
(dlGo to the CM fume. in which each gluon has energy E; express all the kin�matic
factors in terms of E and the scattering angle lI. Add the amplitudes to get the
total. A.
(e) Find the differential scattering cross section.
If) Determine whewr the force is attractive or repulsive (ifit is the former. this may be
i likely glueNll configuration).
9
Weak Interactions
This cho.pler survqs the �MOry of weak interactions. It relies heavily on Chapter 7,
but not on Chaptu 8; Stelio" 4.4.1 would be useful background. 1 begin by staling
lhe Ftynman Rul� for the coupling of leptons to w±, and treat lhru classic problems
in some detail: bela decays of lhe muon, Ike neutron, and the charged pion. Next,
\tit: consider the coupling o
f quarks to W±. which brings in the Cabibbo angle, the
GIM muhanism, and IfIt Kobayashi-Maskawa matrix. In Suticll 9.6, I state the
Feymn('111 rolesfor coupling quarks and leptons to the z!J, and the final stetion sutcks
the G/ashow-Weinbtrg-Sa/am dectroweak thtory. Throughout this chapur I tau the
neutrinos to be massltss; nont oflhe rtsults an" measurably affictld if(minute) neutrino
masstS are inciudtd.
'.1
Charged leptonic: Weak Interactions
in QeD) are the W's (w+ and W-) and the z!l. Unlike the photon and glu.
The mediators or weak interactions (analogous to photons in QED and gluons
ons, which are massless, these 'intermediate vector hosons' are extremely heavy;
experimentally,
Mw = 80.40 ± .03 GeV/,2,
Mz '" 91.188 ± JJ02 GeV/,2
(9.1)
-I), whereas a free massless particle has only two (if z is the direction ofmotion, the
Now, a massive particle of spin 1 has three allowed polarization states (m, ..
I, 0,
'longitudinal' polarization m, .. 0 does not occur). Thus, for photons and gluons,
we imposed both the Lorentz condition
(9.2)
Coulomb gauge ( 0 .. 0, so that (.p = 0, which reduces it further from 3 to 2).
(reducing the number ofindependent components in ,," from 4 to 3) and also the
For the W's and the Z we do not impose the latter constraint. As a result, the
completeness relation is quite different (see Problem 9.1) and the propagator is
In''''''''' '''' t<) Ekm<nlary Partida. St<:ond Edilion. D.vid Griffiths
Copyright e 2008 WllEY·VCH V�rI.g GmbH &Co. KG.A. Weinheim
[SBN: 978·]·527-40601·2
108 1 9
Weak Inrerocrjons
no longer simply -ig.../ q2 , but rather,'
-t(&.. - q..q./Mlcl)
q2
M1cl
(propagator for W and Z)
(9.3)
where M is Mw or Mz, as the case may be. In practice, ql s
i ordinarily so much
smaller than (Me)l that we may safely use
i ".
g
(Mcf
(propagator for q1 « (Me) 1)
(9.4)
However, when a process involves energies that are comparable to Me2 we must,
ofcourse, revert to the exact expression.
The theory of 'charged' weak interactions (mediated by the W's) is simpler than
that for 'neutral' ones (mediated by the 2), so for the moment I shall concentrate
on the former. In this section we consider the coupling of W's to leptons; in the
next section we'll discuss their coupling to quarks and hadrons. The fundamental
leptonic vertex is
1
�
w-
Here an electron, muon, or tau is converted into the associated neutrino, with
emission of a W- (or absorption of W+). The reverse process (v/ .... J- + W+)
is also possible, ofcourse, as well as the 'crossed' reactions involving antileptons.
The Feynman rules are the same as for QED (apart from the modifications already
mentioned to accommodate the massive mediator), except for the vertex factor,
which is
- ...
ig y"(l - y S)
2J2
(weak vertex factor)
(9.S)
The various 2's are purely conventional, and g... = J4:rra... is the ·weak coupling
constant' (analogous to g. in QED and g. in QCD). The term (I - y S ), however, is
of profound importance, for y" alone would represent a vutor coupling (like QED
or QCD), whereas y"y S would be an (lxi(livector (see Equation 7.68). A theory that
•
It might bother you that this dOts not fMuce
to the photon prol"'gator as M ..... O. For par.
ticles of spin 1 (or hightr). the massless limit
is notoriously Irrachtrous. bK.ause in one
critical respect H is not a continuous proce·
dure. The numhM of degrtts of f�om (that
is. the number ofallowed spin orientations)
drops abruptly from 2, + 1 (for M ¢ 0) 102
(for M _ 0). There ue w.lys of formulating
the thr.,ry that allow a smooth Ir.I.nsilion to M
_ O. but only at the cost ofintroducing spuri·
ous nonphysi l states.
ca
9.1 C"otged lepton;'; Weok Interaction'
adds a vt:ctor to an axial vector is bound to violate the conservation of parity, and
this is precisely what happens in the weak interactions:
Example 9.1
Inverse Mvon De,a� Consider the process
represented (in lowest order) by the diagram
�
;':?
"
.
•
ql
"
w
".
� �
Here q_ Pl - Pl, and we'll assumeq2 « M�'z , so we can safely use the simplified
propagator (Equation 9.4); the amplitude is
(9.6)
Applying Casimir's trick (Equation 7.125), and assuming the neutrino masses are
negligible, we find
� I..Al = (8(�:c)2)2 Tr[y'"'(l - YSHil + m.c)Y"(l - YS);ll
.put.
(9.7)
The theorems ofS«tion 7.7 yield
(9.81
for the first trace, and
(9.9)
• In f�CI. the viol�tion is 'maximal'. in th�
s�nse th�t tile two terms �� eq�lly large.
�n parity violation was first consid�r�.
a factor ofthe form (1 + €y') was uSN!, but
�xperi�nts soon dictitN that . _ -1 (�
Problem 9.3). We can it a ·V-A.· ("vector
minus axial �or') coupling. Fermi's original
theory ofbrta dea.y waS a pu� "",,0' theory
{lih QED). and although others propos�
scabr. pseudoscalor. tensor. or pure �l
couplings, it was not until 1956 that any·
one seriously 'ontempbt� mixing terms of
different parity.
1309
110
I 9 Weak Interactions
for the Se<:ond. II follows that'
'" 1..&'12 = 4
�
<1""0
(-",--)4 (PI . P2)(Pl p.)
.
Mwc
(9.10)
Actually, we want the sum over final spins but the average over initial spins. The
electron has two spin states, but (massless) neutrinos (as we learned in Section 4.6)
have only one (ifyou like, the n
i cident neutrinos are alWQ}'S polarized. since they
only come '[eft.handed'). SO
( )4 (PI P2){Pl
(lAI2) = 2 -",--
.
Mw<
(9.11)
. p.)
If we now go to the eM frame, and negle.::t the mass of the electron
(9.12)
where E is the incident electron (or neutrino) energy. The differential scattering
cross section (Equation 6.47) is isotropic (all scattering angles equally likely)
(9.13)
and the total cross section is
]
'
u = 8� ( � ) neE Ill - (m;;)
[
2
M C
'
'
'
)
'
(9.14)
'.2
Decay ofthe Muon
Electron-neutrino scattering is not the easiest thing in the world to study exper·
imentally. but the closely related process, muon decay (J.l. -> t + v., + ii.), is the
cleanest of all weak interaction phenomena, theoretically and experimentally. The
Feynman diagram
�
"
ql �
W p,
� �
• Note th�t '�'''' '�''' ,. -2(6;5',' - &;&:) (Probl.m 7.35). The I�ces in Equ�tion 9.7 �re speci�l
result
ases of � structure th�t wilt OCCUr repe�tedly in this ch�pter, it might k � good idea to pau,,",
hne and wor\:: oul the g�ric
(Problem 9.2).
9.2 Decoy oflhe Muon
leads to the amplitude
(9.15)
from which we obtain, as before,
(9.16)
In the muon rest frame, PI _ (m"c, 0), we have
(9.17)
and since PI - P2 + Pl + p.
(P1 +p.)2 = pi+p! + 2pl . p. = m�cl + 2P1 . p.
= (PI _ 1'2)2 = p� + p� - 2pI . 1'2 = m!? - 2pI . 1'2
(9.18)
from which it follows that
Pl · P. "'"
(m�
_
m:)cl
2
- m" E2
(9.19)
The algebra will be simpler later on, at no significant cost in accuracy, if we set
m, 0, so that
_
(9.20)
(
)(
The decayrate is given by Equation 6.21:*
_
dr -
OAfll)
2!im"
d1p2
(2rr)J2Ipzl
d1 p1
(lrr)J2IPJI
)(
dJp.
(lrr)J2Ip.1
)
x (2rr}"�·(pl - 1'2 - P1 - p.)
(9.21)
To begin with, we peel apart the delta function:
and perform the Pl integral:
(9.23)
• Note !h�1 !his is � 110m body deuy, so w.. haW! 10 go all the way Inck \(I 1M Golden Rule.
1 311
Next we'll do the P2 integraL Setting the polar axis along P� (which isfixed, for the
purposes of the P2 integration), we have
(9.24)
,nd
(9.25)
The � integral is trivial (fd¢ = 2n-); to carry out the 9 integration we change
variables (8 -+ II):
(9.26)
2 1l du = -21P2l lp�1 sin9 d9
(9.27)
where
(9.28)
The u integral is 1 if
(9.29)
1
(and o otherwise) - which is to say (Problem 9.4).
1 +
( lPlI
1 ,, 1 < 1 m"
Ip�1 < !rn"c
Ip41) > ! m"c
(9.30)
These constraints make good sense kinematically: particle 2. for example. gets the
maximum possible momentum when 3 and 4 emerge diametrically opposite to it:
2•
•-
---1'- � 4
In this case 2 picks up half the available energy (� rn!,c2), while 3 and 4 share
the rest. If there is a nonzero angle between 3 and 4, 2 gets less. and 3 plus 4
correspondingly more. Thus 1 m"c is the maximum momentum for any individllal
outgoing particle. and the minimum total for any pair.
The 9 and ¢ integrals have left us with
(9.31)
9.2 �CoyoflheM"o"
The inequalities in Equation 9.30 specify the limits on the IPzl and Ip41 integrals:
Ipzl runs from ml' c - lp41 up to �ml'c, and Ip41 will then go from 0 to ! ml'c.
Putting in Equation 9.20' and carrying out the IP21 integral, we have
�
(9.31)
Finally, writing
( g. )'
and expressing the answer in terms ofthe electron energy. E - Ip.lc, we concludet
dr
dE = Mwc
m;E'
2h(4JT)J
( 4E )
1
- 3m,,'!
(9.33)
This tells us the energy distribution of the electrons emitted n
i muon decay; it
nicely matches the experimental spectrum (Figure 9.1). The total decay rate is
(9.34)
and hence the lifetime ofthe muon is
(9.35)
Notice that g... and Mw do not appear separatdy, either in the muon lifetime
formula or in the electron-neutrino scattering cross section; only their ratio occurs.
It is traditional, in fact, to express weak interaction formulas in terms ofthe 'Fermi
coupling constant'
(9.36)
• Notice that ( I ...NII ) depends only on the m.>grIilli/k of PI. not on it. <Wtaicn: that·s why I waS
frtt 10 ignore it in the {} and 4> integrations.
t Remember Ibat Equation 9.33 applies only up to E "" i"'�? (Equation 9.30). at which point it
drops abruptly to zero (lbe comers are softened a bit by Ibe indusion of p;irtide masses and
tadiati� corrections).
1 313
314
1 9 W�Gk InUrGcfiQns
r-r-'-.--,-r-'-'�r-r-�
�
1 5 X 11)l
>
•
•
�
N
�
0
i. 10 X 103
"
,
•
�
!
z
E
,
5 X 103
o
10
20
Fig. 9.1 Experimental spectrum of po$itron�
in 14+ -+ e+ + v, + �� . The solid line is
the theoretic:.lly predicted spectrum based
on Equation (9.33), corrected for electro·
magnetk effects. (Soll",e; Bardon. M. el 01.
30
Positron momentum (MeV/c)
50
(1965) Physicol R£�iew Lm�". 14, 449. For
the latest high.pre.:ision data on milon de·
cay go to the TWIST collaboration web site
at TRIUMF, Vancouver, Be.)
Thus the muon lifetime is written
(9.37)
in Fermi's original theory of beta decay (1933) there was no W; the interaction
was supposed to be a direct four.particle coupling, represented in the Feynman
language by a diagram ofthe form
�
�
9.3 Der;uYoflh�N�lJlror!
From the modern perspective, Fermi's theory combined the W propagator with
the two vertex factors, in the diagram
,
w
'.
to make an effective four-particle coupling (Onstant Gf. It worked, but only because
the W s
i so heavy that Equation 9.4 is a good approximation to the true propagator
(Equation 9.3)," and n
i fact it was re.-:ognized already n
i the 1950s that Fermi's
theory could not be valid at high energies. The idea of a weak mediator (analogous
to the photon) was suggested by Klein as far back as 1938.
Ifwe put in the observed muon lifetime and mass, we find that
GF/(/ic}l ::
.;; ( Mg;,2 r 1.166 iO-}/GeV1
=
x
(9.38)
The corresponding value of gw s
i
8w =
0.653
(9.39)
and hence the 'weak fine structure constant' is
a.. :: 411".: = 29.51
(9.40)
This number shouldcome as something ofa shock: it is larger than the ele.-:tromag.
netic fine structure constant (a = rt,). by a factor of nearly 51 Weak interactions
are feeble not because the intrinsic coupling is small (it isn't), but because the
mediators are so massive - or, more precisely, because we typically work at energies
so far below the W mass that the denominator in the propagator Iq2 - M�,ll is
extremely large.
'.3
Decay ofthe Neutron
The success of the muon decay formula (Equation 9.33) encourages us to apply
the same methods to the decay of the neutron. n -+ p + e + 'ii,. Of course, the
neutron and proton are composite particles, but just as the Mott and Rutherford
cross sections (whiCh treat the proton as an elementary 'Dirac' particle) give a
• F�rmi �lso thought th� coupling w�s pure
1I«tor, �s [ m�ntionN earU�r. �pjle thrse
defects (for which Fermi could scarcely be
blamN; after aU, he invented the theory
at a time when th� neutrino was a wild
speculation and the Dil7rC equation itself waS
br.lnd new), Fermi's theory was ..tonWtingly
prescient. and all subsequent developments
have been relatively small adjmtments to it.
1315
3161 9 Wrok Inreraaions
good account oflow.energy electron-proton scattering, so we might hope that the
diagram
»e
,1 wi$o
n
�
�
(the same as for muon decay, only with n -0. P + W- in place ofj.t -0. ul' + W-) will
afford a reasonable approximation to neutron beta decay. From a calculational point
ofview the only new feature is that 3 s
i now a massive particle (a proton, instead of
a neutrino). As it happens (Problem 9.8) this does not change the amplitude:
(IAI2) 2 (!J.
=
(PI . J12}(Pl
.
(9.41)
p.)
- same as Equation 9.16. In the rest frame ofthe neutron, we find
J m2 m2 m2 2 CIP 1 )
(JAI2) :: c (�
Mill )"IP2 (
m..
�
_
P
_
,
_
m..
2
(9.42)
But because the electron rest energy is a substantial fraction of the total energy
released,
(mn -mp - m,)c2,
we ClUlIlOJ afford to ignore the electron mass, this time.
lbe decay rate calculation proceeds as before (with the masses now included):
dr
�
x
(;:2:1 C2"�:�;P'I)
(2Jr)\5"(PJ -
P2
C"I'2J�' m;<') C"I'2J�'+ m�" )
- Pl - p")
+
(9.43)
(9.44)
where
(9.45)
To carry out the P2 integral, we again set
(9.46)
9.J �roy of1M Nelllfa"
and orient the coordinates so that the z axis lies along P. (which is fixed, for
purposes of the P2 integral); then
(9.47)
,nd
udu = -IPzllp.) sin9 d9
(9.48)
The ¢ and 9 (or rather, u) integrals yield
dr
�
�
where
d' p
dlp2l l
+
(4JT) Iim Ip.l Pi + m�c2
(IAtll)
4
�
j
jIP4Il + m�cl - u) du
ifu_ « m"c - IP11 - .,IIP412+ m]c2) < I4
1 "" 1:+ (
� 1 1'
� m�c - IP21 -
othelWise
o.
(9.491
)
(9.50)
and the limits are
(9.51)
As before, Equation 9.50 defines the range ofthe Ip21 integral; I'll let you work out
the algebra (Problem 9.9):
p± =
tIm! - m; + m:)c2 - m"Jlp.11+ m�cl
m�c
Jlp412+ m:Cl =F Ip.1
(9.52)
With ( IAtI! ) from Equation 9.42, the 1)):21 integral becomes
(9.53)
and since
(9.54)
we conclude that
dr
dE
1 ( g. )'
= Iic2(4JT) J
Mwc
j(E)
where E = C.,llP41l+ m!Cl is the electron energy.
(9.55)
1 117
318
I 9 Weak InferaaiotlS
E(MeV)
Fig. 9.2 Eleoron en�'gy di�1ribu!ion (,om neutron bet� de·
u�. (Solid line is the theoretic�1 curve: dots ire experimen·
ul diU.) (Sou.u: Christensen, C. J. tf al. (1972) Ph)'S;wl
Rt�jtw. 05, 1628. Figure (9.4).)
Equation 9.55 is =t (use it, ifyou like, to rederive Equation 9.33, by setting mn
-+ m" and m" m, -+ 0). butJ(E) is a rather cumbersome function:
(9.56)
It pays to approximate, at this stage. recognizing that there are four small numbers
here:
f
mn - my
== --- = 0.0014.
m.
E
,- -2
m"c
Ii E "" = 0.0005.
m.
(� < I) < f).
<$> == l.&!
m.,
(9.57)
(0 < <$> < 1)
(The last of these is not independent. of course: <$>2 _ 1)2 - �2 .) Expanding to lowest
order (Problem 9.9), we obtain
(9.58)
So the distribution of electron energies is given by
dr
-
dE
�
1
-
(
) ./
..
'
-£
]flti. 2Mw,1
£2
_
m2c'
. [(m" - I'np)cl- E)
'
(9.59)
The experimental results are shown in Figure 9.2. The electron energies range
from m,,2 up to about (m" - mp),2 (Problem 9.10). Integrating over E, we get the
9.3 Decoy offh� N�ufrol1
total decay rate (Problem
r=
4JTl,'i
'
(2Mw,1 ) (m,' )
""
'
"
[ 1s(2a. - 9a2 - 8) � + aln(a + �]
1
x
where
9.11):
a m,, - mp
..
(9.61)
m.
Putting in the numbers, I find (Problem
T =
r=
,
(9.60)
9,12)
1 3 18 s
(9.62)
This is in the ball park, as they say: the experimental neutron lifetime' is 885.7
± 0.8 seconds. and given that weak decays range from IS minutes down to
10�lJ
seconds we should perhaps be satisfied to get the right order of magnitude. But
why s
i n't the agreement better?
The main problem is that we have treated the proton and neutron as though they
were simple point particles, interacting with the W in exacdy the same way leptons
do. To be honest about it, we should go back to the beginning, admit that we do
not really know how the W couples to composite structures, draw n
i a blob on the
Feynman diagram (to symbolize our ignorance)
,
".
"
p
and express the amplitude in terms of various unknown 'form factors', whose
structure is limited only by Lorentz covariance - just as we did in Chapter 8 for
the proton-phatcn vertex. Not until a mature QCD can provide us with the detailed
structure of the nucleons will we be in a position to perfect the neutron lifetime
calculation.
• This numhrr is from tht 2006 Particle Physics
S""Jdtl IPPB). Fr� ntulron. art hard 10 work
with, and th� 'official' ntulrOn lifetim� has
changM substanliaUy over th� years (th� firsl
PPB listed it as 10«1 ::I: BO s«onds). Note
also that nucl�ar physicisl$ tend 10 quote tht
""lf�ifo (I" l � r 1n2), and hrta..J=oy sp«ial·
isl$ often quote th� 'comparative half·lif�' -
thr ....calkd 'ft' valur - which has ",nain
kin�matic and Coulombic contributions r�·
moved (for th� n�utron th� corre.:tion faclor is about 1.7). This is jw;t to warn you that
th� numhrrs giv�n in tilt lit=tur� for th�
nrutron 'lifttimr' art aU over tht map, and it
pays to r�ad thr fine print and <bed th� date.
J 119
)20
I 9 Weak Interaction!
And yet, the Mott formula works well for low·energy electron-proton scattering:
why does essentially the same procedure give us the right answer in electrodynam­
ics, but not in the weak interactions? In both cases the wavelength of the 'probe'
(y or W, as the case may be) is much larger than the diameter of the 'target'
(p or n) (see Problem 9.13); the nucleon's internal structure is not 'resolved', and it
behaves as a point particle. The crucial question, though, is: what is the net coupling
stnngth of this object? Of course, the net charge of the proton is still e - it doesn't
matter what complicated processes are going on inside - valence quarks emitting
virtual gluons, gluons producing quark-antiquark pairs, 'sea' quarks recombining,
and so on - because aU /hisfrenzied activity conserves charge. From the perspective
of a long wavelength photon it just looks like a point, and the net charge of the
composite nucleon is just the sum of the charges of the valence quarks. But there
is no a priori reason to suppose that the same applies to the weak coupling; when
a gluon splits into a quark-antiquark pair, the net contribution of this pair to the
weak coupling may not be zero - who knows? To account for this, we make the
following replacement in the n """"" p + W vertex factor:
(9.63)
where Cy is the correction to the vector 'weak charge', and 'A is the correction
to the axial vector 'weak charge', Luckily, the same basic process, n """"" p + e + V"
occurs not only for the fra neutron, but also within radioactive nuclei, so we
have, in principle, many independent opportunities to measure Cy and cA." The
experimental results are as follows:
Cv ==
1.000,
CA =
1.270 ± 0.003
(9.64)
Surprisingly, the vectorweakcharge is not modified by the strong interactions within
the nucleon. Presumably, like electric charge, it is 'prote<ted' by a conservation law;
we call this the 'Conserved Vector Current' (CVC) hypothesis.t Even the axial term
is not altered much; evidently it is 'almost' conserved. We call this the 'Partially
Conserved Axial Current' (PCAC) hypothesis.
The effect of this substitution (Equation 9.63) on the neutron lifetime is some·
thing you can calculate for yourself, ifyou havethestamina; to good approximation,
the de<ay rate is increased by a factor of
(9.65)
and the lifetime is decreased in the same ratio:
,�
1316 s
= 901 s
lA6
(9.66)
--
• A �rticular favorite is 1'0
"N, which is known 1from the obsel"'led spin and parity of the
initial and final st-.tes) to involve only ""tor coupling.
t eve is built into the Standard Model, and nowldays 'V is taken to � 1 exacdy: the experiments
are interpreted as melSurements of the Cabibbo angle (see below) - or, more predsely of V...
->
,
9.4 Decay offhe Pion
This is now within striking distance of the experimental value. Unfortunately,
the agreement s
i deceptive, for there is yet "noiliu correction to be made. The
underlying quark process here is d ...... u + W (with two spectators):
•
".
w
i the 'Cabibbo
and this quark vertex carries a factor of cos 8e, where 8e 13.150 s
angle'. I'll have more to say about this in Section 9.5, but the essential point for now
is that our theoretical value for the neutron lifetime, corrected for nonconservation
ofthe axial charge and modified by the Cabibbo angle, is
_
90"
cos 8e
,= = 950s
,
(9.67)
Two steps forward, one step back!'
,..
Decay ofthe Pion
According to the quark model, the decay of a charged pion (tr- ...... 1- + VI, where
I is a muon or an electron) is really a seaturing event in which the incident quarks
happen to be bound together:
w
In this sense, it is a weak interaction analog to positronium decay (t+ + �- ...... y +
y) or "Ie decay (e + c ...... g + g) - electromagnetic and strong processes, respectively.
We could analyze it this way, following the methods of Example 7.8 and Section 8.5
(see Problem 9.14), but in the end we would be stuck with a factor ofl1/t(0)11, andat
this stage we have no idea what the wave function (1/t) ofthe quarks within a pion
• "This isn't th� �nd of th� story; th�re is, for enmple. a snull Coulomb correction, (due to the at·
traction of the electron and proton in the final sb.�), But _ are within 7% of th� aperm
i enb.l
result. and it is time to move on.
j 121
3121 9 Wowk inluQrtio'"
looks like. Given that such a calculation will carry this undetermined multiplicative
factor anyway, it is simpler to proceed as follows.
Redraw the Feynman diagram, with a blob to represent the coupling oflr - to w-:
p
w
/'I,
We may not know how the W couples to the pion, but we do know how it couples
to the leptons, so the amplitude must have the general form
,
.At = �[ii(3)y,,(l
8(Mwc)
- yS)u(2W"
(9.68)
where F" is a 'form factor' describing the 11" -+ W blob. It has to bea four·vector, to
contract with the y" in the lepton term. But the pion has spin zero; the only vector
associated with it, out of which we might construct F", is its momentum, pI' .• (I
won't bother with a subscript on the pion's momentum: P ""! Pl') So F" must be
some scalar quantity times pI':t
(9.69)
In principle.j" is a function of f - the only available scalar - but since the pion
is on its mass shell cr m!c1).jw is, for our purposes, a fixed number, the 'pion
decay constant'.:
Summing over the outgoing spins, we get
=
(I.AtI') = [4 (�c)']'P"P>Tr[y"(l - YS)f2y"(l - Y S)flJ ffljC)]
=
��� (�:J r
2
+
[2(p . PI)(P ' pJ) -I!P2
• Notice that _ inuoduce the (�al<) pion
form factor at thr lr""l of .A, whereas for
the (electromagnetic) proton form factou �
waited until the ( I..kll) stage. Thr rrason
is that thr proton hu a spin. and we would
ha"" to include thai in the ros,",r of availabk
vectors: it is only aflM wr have """'age4 over
the spins that the list reduces to two. md
the problem beeo""'s numgeable. Thr pion.
ho�, has no spin, so we can afford to in­
troduce the form faclor directly in ..k, where
it is only a l'tctor quantity. instead of a len·
w<
PJ))
(9.70)
t For reasons that will appear in the next
section. it is customary nowadays 10 factor out
the appropria,", Cabibbo-Kob,ayashi-Maskawa
(CKM) matrix ekment in the drfinition of
the "",son decay <Qnstants:.r. -. Vo• J.- TQ
a""id cluttered MlatiQn 1"11 use the Qlder
(on""ntiQn.
t lbe CQrresp<>nding faclor fQr Qther pseu.
ooscalar mesons will involve a different value
Qf r. and a different element in the CKM
matrix (S« foomQtetl·
'14
Dewy ofthe Pion
(the trace was already calculated in Equation 9.8). But P - P2 +Pl, so
(9.71)
p . P2 = P2 . P3,
(9.72)
Thus
(9.73)
(a constallt).
The decay rate s
i given by the standard formula (Equation 9.35):
(9.74)
and the outgoing momentum is (see Equation 9.34 or Problem 3.19)
(9.75)
So
(9.76)
Of course, without knowing the decay constant,!", we cannot calculate the pion
lifetime: But we are able to determine the ratio of the electron and muon decay
rates:
(9.77)
(The experimental number is 1.230 ± 0.004 x 10-4.) At first glance, this is a very
surprising result, for it predicts (correctly) that the pion prefers the muoll mode,
in spite of the fact that the electron is much lighter. phase space considerations
favor decays for which the mass decrease is as large as possible, and unless some
conservation law intervenes, we ordinari
l y find that the lightest final state is the
most common one. Pion decay is the notorious exception, and it calls for some
special dynamical explanation. A due is suggeste<l by Equation 9.76: notice that if
the electron were massless, the ]f - -+ e- + ii, mode would be forbidden altogethtr.
• It is � r�ther strilcins f�ct th�t ifyou put in!. m.� (or, b,mer�, m.e cos /Ie) you 'Orne out
very close to the ,..- .... ",- +Ii� lifetime. but I know of no persWisive theoretial justification
for this ansalZ. and it doesn·t II/(Irk for the heav;"r mesons.
_
1 121
3241 9 Weak Inleractians
Can we understand this limiting case? Yes: the pion has spin 0, so the electron and
the antineutrino must emerge with opposite spins, and hence equal helicities:
1
The antineutrino is always right-handed, so the electron must be right.handed as
well. But if the electron were truly massless, then (like the neutrino) it would only
exist as a left·handed particle. More precisely, the
-
y S in the weak vertex factor
would couple only to left-handed electrons, just as it couples only to left·handed
rr- -jo e- + v, could not occur at aU, and why (the physical electron being very clo�
neutrinos (see Problem 9.15). That's why, if the electron were massless, the decay
to massless) the decay is so heavily suppressed.
•.,
Charged Weak Interactions ofQuarks
In the case of leptons, the coupling to W±takes place strictly within a particular
generation:
(lepton generations)
v. + W-, J.L- -+ v" + W-, r - -jo II, + W-, but there is no
cross·generational coupling, ofthe form e- -jo v" + W-, for example. The wup!ing
of W to quarks is not quite so simple, for although the generation structure is
That is, e- -jo
similar
(�). (:). (:)
(quark generations)
the weak interactions do not strictly respect their separate identities. There are, to
be sure, interactions of the form d -jo u + W- (the process that underlies neutron
decay, n -jo p + e + v,), but there exist as well cross·generational couplings, such as
s -jo u + w- (seen, for example, in the decay A -jo P + e + vt). Indeed, ifthis were
not the case, we would have three absolute 'flavor-(onservation' laws: conservation
of 'upness·plus·downness', 'charm.plus·strangeness', and 'truth·plus·beauty' analogous to the three lepton number conservation laws. The lightest strange
particle
(K-) would be absolutely stable, and so would the B meson (the lightest
beautiful particle); our world would be a quite different place.
In 1963 (when u, d, and s were the only quarks known), Cabibbo (lJ suggested
that the d -jo It + W- vertex carries a factor of cos {Jc, whereas s -jo It + W-
9.5 Charged w""k /nuraajans ofQuarlr..
carries a factor ofsin {Je; apart from that they are identical to the leptonic couplings
(Equation 9.5):
1
1
� �
w-
w-
(9.78)
The strangeness-changing process (s -.. u + W-) is conspicuously weaker than the
strangeness-conserving one (d -.. u + W-), so evidently the 'Cabibbo angle' Be is
rather small. Experimentally,
(Je = 13.15"
(9.79)
The weak interactions almost respect quark generations . . . but not quite.
Example 9.2 Leptonic �coys Consider the decay K- -.. 1- + ii), where I is an
electron or a muon. This is the analog to ]f - decay (Section 9.4), but now the quark
vertex is s + U -.. W-, instead of d + u -.. W-. From Equation 9.76 we have
The coupling strength is presumably about the same, except that where J"
contained a factor of cos Be.JK carries a factor of sin {Je. Accordingly,
(9.80)
Putting in the numbers, I get 0.96 for the muon mode (I - �) and 0.19 for the
electron mode (I _ e). The absented ratios are 1.34 and 0.26, respectively; these
decays are pure axial vector, and, as we discovered earlier (Section 9.3), perfect
agreement is not to be expected. _
Processes ofthe kind considered in Example 9.2 are called Itptonic decays. There
also exist semiltptonic decays, such as ]f - -.. ]fo + e- + ii., �
K -" ]f+ + �- + iiI'
(Figure 9.3a), or for that matter the beta decay of the neutron: n -.. p+ + e- + ii•.
Finally, there are nonltptonic weak interactions, such as K- -.. ]f0 + ]f- or A -..
p+ + ]f- (Figure 9.3b). Generally speaking, the latter are the hardest to analyze,
be<ause there is strong interaction contamination at both ends of the W line [2).
1 325
d
(R'1
,
"
w
w
,
"
:::::.-----:: ('It +)
d
(A)
d
�(P)
(bj
Fig. 9.3 (�) A typic�1 semileplonic dec�y (it ..... If+ + /.1 +
i!�). (b) A typical nonleplonic wuk dtcay (A ..... p. + .!I'-).
,.j
-
Example 9.3 Semileptanic Decays In the case of neutron decay (n --;. p + e + ii,j.
the basic quark process is d ---+ u + W- (with two spectators). However, there
are two d quarks in the neutron. and eilMr OIU could couple to the W; the
net amplitude for the process is the sum. The simplest way to keep track of
the numbers s
i to use the quark wave functions in Section 5.6.1; the Aavor
states '/1n. for instance, give n = (ud - duj dj.;2, from which (with d --+ 1.1) we
get ((1.11.1 - uu)d + (ud - du)ul/...!2 = (ud - du)u/v'2 = p. The overall coefficient is
then simply (as Be (as [ claimed at the end of Section 9.3). By contrast, in the decay
1:0 --+ 1:+ + e + ii,. the quark process is still d -+ 1.1, but here EO .::::: [(us - su)d +
(ds - sd)uJf2 ...... [(us - su)u + {us - su)uJf2 = (us - su)u = J2:E+, and hence the
amplitude carries a factor of J2 cos 00'- The decay rate is given by Equation 9.60,
which reduces (in the case,, » 1) to the fonn
where lI.m is the baryon mass decrease and X is the Cabibbo factor (cos Oc, for
neutron decay; J2 cosBc, for :Eo ...... r:+ + e + ii.; etc.). I'll let you work out the
numbers for yourself (Problem 9.17).t U
Cabibbo's theory was very successful in correlating dozens of decay rates, but
there remained a disturbing problem: this picture allowed the J('J to decay into a
p.+p.- pair (see Figure 9.4). The amplitude should be proportional to sin Oc cos
Be, but the calculated rate was far greater than the experimental limit. A solution
• Acrually. tMr� is a technical diff�r�nc� h�r�: tM acti� quark is bound 10 th� spectator in a .pi"
sirtgla state. Fortunately. this does not affect the Hfetim�.
t This procedur� includes only th� Ya\enc� quarks. and h�nc� is insensitive 10 th� nonconser·
vation of th� axial coupling. As we found in Equation 9.65, PCAC can l�ad to a correction of
nearly SQ%, 50 on� does not expect fine precision in the lifetimes. Cahihbo's theory included a
way of calculating the axial couplings, but ! shall not go into that h�r�.
K
)
,
•
W
"
'
d
'.
K
W
•
Fig, 9.4 The dec�y KO ---> .' + ...- .
)
9.S Chorged weak /nUroaian, afQINJrf<
•
,
•
W
,
'
d
'.
W
•
Fig. 9,S The GIM mechanism. This diigr�m
cancels Figure 9A. Note the virtual , quark
replilong the u.
to this dilemma was proposed in 1970 by Glashow, Iliopoulos, and Maiani (GIM)
[3}. They introduced a fourth quark, c (note that this was four years before the
'November Revolution' produced the first dire<:t e:q>erimental evidence for charm)
whose couplings to s and d carry factors of cos Be and - sin Be, respectively:
l
l
� �
w-
w-
� yl'( l - y>)(- sinBc)
(9.81)
-
2v2
In the 'GIM me<:hanism', the diagram in Figure 9.4 is canceled by thecorresponding
diagram with c in place of u (Figure 9.5), for this rime the amplitude is proportional
to - sin Be cos Be"
The Cabibbo-GIM scheme invites a simple and beautiful interpretation: instead
of the physical quarks d and 5, the 'correct' states to use in the weak interactions
are tf and i, given by
cl = dcosBe + ssin8e,
s' = -dsinBc + s cosBe
• The clncel�tion is not P'tftd, beau� the
mass of the e is not the ume as the IN.SS
of the ... Howe""r, these virtrnll !"Irticles are
SO far off the mass shell that both pr0!"lga·
tors art essentially just ;�/<i. (In Glkulating
AI: we sholl be in�ratin8 over the one rt·
maining in�rnal momenNm which is not
fuce<l by the conservation laws. n,is is essen·
tUlly the momenNm 'circulating around the
(9.82)
loop'. llecauSO!' of the two W propagators, the
main contribution will co� in the region of
the W mass, which is so much greater than
the , or " mass that the lallet can, to good ap­
proximation, be neglected. Actually, the de·
uy WS <xcur, it's just elCtremely slow, and if
you ilLdlld< the effects of w/e mass difference,
the calculation is consis�nt with the ob��
rate,)
1 321
32S I 9 Weak 'nt.,aaion5
or, in matrix form
( )= (
) (d)
d'
"" oe
,'nBC
1
- sinOc
cosBc
S
(9,83)
The W's couple to the 'Cabibbo-rotated' states
in exactly the same way that they couple to lepton pairs,
(�.) (�);
and
their
couplings to the physical particles (states of specific Aavor) are then given by
(9.84)
That is, d _ l.( + W- carries a factor cos Oc, S --+ l.( + W- carries a factor sin Oc,
and so on:
At the time, the GIM mechanism seemed a little extravagant - introducing a
new quark just to fix a rather esoteric technical defect in a largely untested theory.
But the skeptics were silenced by the discovery of the .p(c') in 1974. Meanwhile,
Kobayashi and Maskawa [41 had generalized the Cabibbo-GIM scheme to handle
three generations of quarks.t the 'weak n
i teraction generations',
(9.8S)
"",ale into':ractions and we need to rOlate back
to get the 'physical' stato':s (see Chapter 11).
t It is interesting to note that Kobayashi and
th� ";"n� purpose by introducing " _ u cos
Masbwa proposed a third quark generation
9c - c sin ge and t:' _ u sin ge + c cos ge.
before the ="d waS complete, and long be­
Incidentally, you might be wondering whether
fore there was any experimental eviden� for
a similar rotation occurs in the ItpIOIi sec·
.. third. They were moti....ted by a desire to
tor. If aU neutrinos wele massless, any linear
expwn CP violation within the c..bibbo-GIM
combination of them would <lilI be massl....,
scheme. It turned out th..t for this purpose
and there would be no 'tag' to identify the
they needed a complex number in the 'rota·
'umot�t«l' stat.... But, f
i neutrinos ha� mass
lion' matrix (Equation 9.83), but such a term
(... _ now know they do), ther� is no rea·
could a1w..ys be eliminated by suitable ,«le{.
son to suppose that the 'mass eigenstato':s' ar�
inilion of the quark phases, unl.,,;,; tMY went
the ...
me as the weak imeraction states, ..nd
to l 3 x 3 matrix, lnd hence to three generl'
the """e rotation story plays out _ only in reo
lions (Problem 9.18).
....=, since the 'familiar' nrutrinos are the
on... paired with the charg«lleprons in the
• It is pur�ly con�ntional that w� 'rotato':' d and
5, rather than u and c; we could accomplish
9.6 Ne�lrol Weak /nlerodions
are related to the physical quark states by the CKM matrix:
(9.86)
where V"-<I, for example, specifies the coupling of u to d (d -4 u + W- ).
lbere are nine (complex) elements n
i the CKM matrix, but they are not all
independent (see Problem 9.18); V can be reduced to a kind of 'canonical form', in
which there remain just three 'generalized Cabibbo angles', (0\2, 02], 00) and one
phase factor (�) [SJ :
S\2Cll
CllCn - SUSllsnt,
-C12Sl] - S\2C2lsUd'
)
SHCll
C2lCll
''''-'
(9.87)
Here cij stands for cos Oij, and sij for sin Oij. If On _ Oil _ 0, the third generation
does not mix with the other two, and we recover the Cabibbo-GIM picture, with
B12 - Oe. However, there is compelling evidence (namely, the observed deGIY of
the B-Wu) meson) for some third·generation mixing, although it must be fairly
small in order to account for the success of the original Cabibbo-GIM scheme.
The Standard Model offers no insight into the CKM matrix (indeed. this is one of
its most conspicuous weaknesses); for the moment, we simply take the values of
the matrix elements from experiment. The magnitudes are [6]:
IVijl ""
•••
(
0.9738
0.2271
0.0081
0.2272
0.9730
0.0416
Neutral Weak Interactions
)
0.0040
0.0422
0.9991
(9.88)
In 1958, Bludman [7] suggested that there might exist lUulral weak interactions,
mediated by an uncharged partner ofthe W's - the zfJ:
Here f stands for any lepton or any quark. Notice that the same fermion comes
out as went in (just as in QED and QCD). We do not allow couplings of the form
J,L- -.. e- + zfJ, for example (this would violate conservation of muon and electron
number), nor ofthe form S -4 d + zfJ (such a strangeness-changing neutral process
would lead to � -4 /1-+ + /1--, which, as I have already remarked, s
i strongly
1329
"
,­
:,1 "-,
fig. 9.6 The �rst picture of a neutral weak
pr<><:ess [iiI' + e- ..... li,. +e-). The neutrino
enters from below �eaving no trad), and
strikes an electron, which moves off (up­
ward), emitting two photons (which show
up in the �gure only when they subsequently
produce electron-positron plirs) as it slows
down and spirils inward in the superim­
posed magnetic �eld (the big circle in me
lower left is a lamp)_ {Sou",e: (ERN.)
suppressed).' [n 1961, Glashow [8J published the first paper on unification ofweak
and electromagnetic interactions; his theory required the existence of neutral weak
processes, and specified their strocture (see Section 9.7). In 1%7, Weinberg and
Salam [9J formulated Glashow's model as a 'spontaneously broken gauge theory',
and in 1971, 't Hooft (10) demonstrated that theGlashow-Weinberg-Salam(GWS)
scheme is renormalizable. Thus there were increasingly persuasive theoretical
reasons for thinking that neutral weak interactions occur in nature, but for a long
time there were no experimel1tal data to support this hope. Finally, in 1973 [IlJ,
X'::: :�.;�:::::"::: ::: ;::::::::::
• In the case of newlr4.1 processes, it doesn't matter whether )'Ou use the physical states (d, s, b) Or
�
:
'
'
,
give; .4 - 7:1 = ddsin'9c + «<<>s'9, + (d., + Sd) sin9ccos9c.
So the sum of the two is .4 - ;;-of +- U _ tid .. it. Thus the n<l amplitude, once both di<lgrams
i
is the s:.me whichever states we use. (The s:.me argument gettera)izes to thn.e
generations, as Ion8 as the CKM matrix is unitary.)
are comb ned,
9.6 Ntulrol Wtak Inuroaian,
a bubble chamber photograph at CERN (Figure 9.6) revealed the first convincing
evidence for the reaction
jj,, + e � jj,, + e
suggesting mediation by the ZO;
,
•
z
""
The same series ofexperiments also witnessed the corresponding neutrino-quark
process, in the fonn ofinclusive neutrino�nudeon scattering:
jj,, + N � jj,, + X
v,, + N � v,, + X
Their cross sections were about a third as large as those of the related charged
events (iiI' + N � j.t+ + X and v" + N � j.t- + X), indicating that this was indeed
i her·order process,
a new kind of weak interaction, and not simply the hg
•
w
•
".
w
(which would yield a far smaller cross se<:tion). The CERN results came as welcome
encouragement to eledroweak theorists, who had been out on a limb now for
several years [12].
As we have seen, the coupling of quarks and leptons to w± is a universal 'V·A'
form; the vertex faclor is always
(Wo!< vertex factor)
(9.89)
(It is true that the axial coupling to composite structures, such as the proton. is
modified, but that is a result of strong interaction contamination - the underlying
quark process is pure V.A.) The coupling of the ZO is not so simple:
(ZO vertex factor)
(9.90)
332
1
9 W""k Inuraction,
Table 9.1 Neutril Vedor ind ixiil �to' couplings in the CWS model
"
f
�.. v". v,
"
e-. /A . '
- t + 2sinl0",
d. s. b
_ i + � sinlo...
". C.I
i _ j sin1o...
,
,
-j
j
- ,,
where gz is the neutral coupling constant. and the coefficients
4. and � depend on
determined by a single fundamental parameter 0..,. called the 'weak mixing angle'
the particular quark or lepton if) involved. In the GWS model, aU these numbers are
(or 'Weinberg angle'), as indicated in Table 9.1. The weak and electromagnetic
coupling constants are related:
g, = sin8..g.cos8...
(9.91)
where St, remember, is essentially the charge of the electron (g.. = e./4if7lfC).
Finally, the W�and ZO masses are related by
Mw = Mz(Qs8..
(9.92)
Equations 9.9{}-9.92 are the basic predictions of the GWS model; you'll see how
they were obtained in the next set:tion.
The Standard Model provides no way to calculate 8,.. itself; like the CKM matrix,
its value is taken from experiment:
(sin1 8.. = 0.2314)
(9.93)
Butgiven the value of8..., we can calculate the W and Z masses (see Problem 9.20).
Their discovery by Rubbia at CERN in 1983, at Mw-82 GeV/ ,1 and Mz _ 92 GeV/
,2 (as predicted) was persuasive evidence for the GWS model [13].
Example 9,4 Elasti' Neufrino-Electron Scattering
In Example 9.1 we calculated the
cross section for the W-mediated process VI' + e __ v. + J.L. We now consider the
related ZO·mediated reaction v" + e __ v" + e.
� �
ql
•
z
",
""
::;: �
9.6 N�lJlro/ W�Qk /nurQdian�
i (Equation 9.3)
The zfJ propagator s
q M�2)
q2 ql'M�cl,/
(f « M�(2)
(&,. -
-i
At low energies
(9.94)
it reduces to
(9.95)
With this appro:timation. the amplitude is
(9.96)
and hence (Problem 9.2)
(1...Ifll) = 2
x
=
(4�Z')·
Trlyl' (l - y5 )lly"(1 - y5)1)]
Tr(y,,(cv - CAYs)qt2 + me)y.(cv - C....yl)qt. + me)]
� (�cr {Icy )2(pl . Pl){Pl P4]
- ,....)l (p) P.)(Pl Pj) - )l � � (P . Pl))
.
+ (cv
+ c....
.
.
(mc (c
-
C ) I
(9.97)
where m is the mass ofthe electron. and Cv and c.... are the neutral weak couplings
for the electron. If we now go to the eM frame. and ignore the electron mass
(m -+ 0). we find
(9.98)
where E is the electron (or neutrino) energy. and () is the scattering angle
(Figure 9.7). The differential scattering cross section (Equation 9.47) is
do
dQ
( ) (4Mzcl ) E, [
"
= 2 -;;
'
g
,
'
,
) cos 2"
c -c " 'J
(cv + c....) + ( v
....
(9.99)
and the total cross section (integrating over aU angles) is
)
(9. 100
1 333
Beton:
Fig. 9,7 EI�stic neutrino-electron scattering in the eM,
Putting in the GWS values for Cv and 'A (from Table 9_1), and comparing the result
of Example 9.1 (Equation 9,14), we find that for energies substantially above the
muon mass
(9.101)
The current experimental value [14] is 0.11, which, given the 10% uncertainties in
the measurements, s
i reasonable agreement. _
You might well ask why ittook so long for neutral weak interactions tobe detected
in the laboratory; after all, 15 years separate Bludman's original speculations from
the definitive experiments at CERN. The reason is that most neutral processes
are 'masked' by competing electromagnetic ones. For example, e+ + e- - ;t+
+ ;t
can occur tither by a virtual z!l or by a virtual y (Figure 9.8); at low
energies the photon mechanism overwhelmingly dominates.' That's why neutrino
scattering was originally used to confirmthe existence of neutral weak interactions;
neutrinos have no electromagnetic coupling, so the weak effe<:ts are not obscured.
But neutrino experiments are notoriously difficult - hence the long delay. An
alternative is to work at high energy - specifically, in the neighborhood of the
ZO mass, where the denominator of the z!l propagator is small, and the 'weak'
interaction is correspondingly large. In the early days it was hard to estimate 0...,
and hence the z!l mass was quite uncertain. But by the late seventies, a variety of
experimental data pointed to Ow "'" 29Q , and hence to Mz 90 GeVIt?- [see Problem
9.20). This prediction was stunningly confirmed in 1983 [131, and inspired the
_
,
,
z
•
Fig. 9.8 Weak �nd electromagnetic contributions to e" + e- ..... ..... + ...-
• In principle, there is weak contamination in
Mry dectromagnetic process, since the z!'
couples to everything the y does {�nd then
some�. For �mple, the Coulomb potential
binding the electrons to the nucleus in an
atom is slightly modifiM by z!' exchange,
and Ihis i. observable n
i atomic spe<I",.
Similarly, there is a weak contribution to
electron-proton SCl.ttering. Although these
effects are minute, they Ie."" • teU.tale
fingerprint: p.rity violation 115\,
q
e.f:1
z
9.6 NeulnJl Weak inU:n:ldions
P/f'
Fig. 9.9 Electron-positron scattering near the ZO pol�.
construction ofelectron-positron colliders designed to operate at the ZO peak: SLC
at SLAC, and LEP at CERN.
Example 9.5 Electron-Positron Scattering Nearthe z!l Pole Consider the process ,+
+ ,- f +j(Figure 9.9), where! isanyquarkor lepton.' This time we shall not use
the approximate form of the z!! propagator (Equation 9.95), for we are interested
precisely in the regime ct <::: (Mzc)2 . The amplitude is
......
(9.102)
where q -PI + P2 -pJ + p•• Since we are working in the vicinity of90 GeV, we can
afford to ignore the lepton and quark masses.t In this case the second term in the
propagator contributes nothing, for q" contracts with y" to give
u(4)j1(cv - CAys)v(3)
but }f = '13 + 'I. and ii(4)'I. "" 0 (Equation 9.96 with m - 0), and
'I*v - CAys)v(3) "" (ev + cAY \') u(3) = 0
for the same reason. Thus
and it follows that
(loL) 2 ) =
[S(f g{MzC)2) ] 2 Tr(y"(4 - �ys)'IJY·(4 _ �yS),.)
x Tr[ y,, (c� - C�yS)'lY"(c'V - C�yS)' J
2
(9.104)
• No: an electron, however. for then we would have to n
i clude th� rotated di"llram.
t I assum� mf « Mz, which rxclud... th� top quark. But th� I cannot br producoNl anyway, at
theu energi....
1 33S
9 Weak InUrIlCliom
Now, the first trace s
i (Problem 9.2)
(9.105)
and there is the corresponding expression for the s«ond trace, so
O.L1 2) =
� [ if f�ZC)2 r 1[(4)2 (c�.l][(C�)2 (cA)2]
+
+
X [(PI · PJ)(P2 . p.) + (PI . P.)\P2 . Pl)]
+ 44�c"vcA[(Pl . PJ)\P2 . p.) - (PI . p.)(Pz PI)]}
•
(9.106)
In the eM frame this reduces to
(9.107)
where E is the energy of each particle and B is the angle between PI and PI- The
differential scattering cross section (Equation 9.47) is, therefore,
(9.108)
and the total cross section is
(9.109)
As it stands, a blows up at the ZO pole - that is, when the total energy (2E) hits
the value Mzc2 (just right to put the ZO on its mass shell). The problem is that we
have treated the ZO as a stable particle, which it s
i not. Its lifetime is finite, and this
has the effect of'smearing out' its mass. We can account for this by modifying the
propagator [16J
I
(Mzc)2 -->-
'q'>=CI'M;:,O'OI'C+'"ih"MO,OrC::,
(9.110)
where rz is the decay rate (experimentally.
adjustment. the cross section be<omes
u =
(!icg:E)2
4811"
�G NtulnJlWtak I"uroaiom
fz 3.791 0.003 1024/s).
-
±
x
[(4l + (c�iH(c�)2 + (CA)2J
[(2 E)2 (MZC2)lJ2 + (IiMzc2rz)1
With this
(9.111)
Because lirz « Mzc2 • the correction for finite z? lifetime is negligible except
in the immediate vicinity of the ZO pole. where it has the effect of softening the
infinite spike.
In Chapter 8 we calculated the cross section for the same process when mediated
by a photon (Equation 8.6J:
(9.112)
(where o!is the charge ofJ. in units of Thus the ratio ofweak to electromagnetic
rates in (for example) muon production. is
e).
u(e+e- z!! JL+JC) = u
u(e+e- --+ Y --+ JL+JL-)
--+
I
--+
-
]2 )
2sin2 /:1", + 4 sin· e...
(sin e... cos e..)4
(9.11 3)
The factor in curly brackets s
i approximately 2. Substantially bdow the z? pole
«
then
Mze2).
•
•,
;;;;: ( E )
uy 2 Mzc2
'
�Mze2.
(2E
(9.114)
and the electromagnetic route dominates (at
=
for instance. the weak
contribution is less than 1%). But right on the z!! pole (2E _ MZ,2).
Uz ;;;;: !
f1y
8
(Mzli C2)
rz 200
""
2E
(9.115)
At the z!! pole. therefore. the weak mechanism is favored. by a factor ofaround
(Figure
_
9.10):
• Equ.ally n
i teresting is the electrom�gneti'·
+ .4"d •
weak 'intrrfrrnlce' that occurs whnl the two
amplitudes ;are combined: 1.4" r
1.4".12 + l.4"d + 2 Re(..Kr ..kz). We h�ve
akulated I"""d and (in Ch�pt.., 8) 1.4".11•
but the cross term provides � sensitive test
of the GWS theory. even at energies subsIan·
ti�lIy below the ZO pole. (See Halzen �nd
200
Martin, ref. (11). SK\ion 13.6. and ref. (ISJ.)
Indeed. it w�s the success of the electroweak
n
i terfrren� experiments n
i 1978 that (On·
vinced most theorists th�t the GWS model
s
i coned. For a contemporary account. see
P"�ia Today. September 1978. p. 17.
H8 1 9 W""k III�rod;olls
«
'"
�
�
"
.
!2
l ro
•
"
"
•
o ,
"
E_(GeV)
Fig. 9.10 Eledron·positron $cattering in the neighborhood of the zC pole.
9.7
Electroweak Unification
9.7.1
Chiral Fl!rmion Statl!s
the cards are now on the table;" it remains only to ex-plain where the GWS
parameters inTable9.1 and Equations 9.90-9.92 come from. Glashow's original aim
was to unifY the weak and electromagnetic interactions -to combine them into a
single theoretical system, in which they wouldappear not as unrelated phenomena,
but rather as different manifestations ofone fundamental 'electroweak' interaction.
was a bold proposition, in 1961 [17J. In the first place. there was the enormous
disparity in strength between weak and electromagnetic forces. However, as
Glashowandothers recognized, this couldbe accounted for ifthe weak interactions
were mediated by ernemely massive particles. Ofcourse, this immediatelybegs the
secondquestion: ifit's reallyall onebasic interaction, howcometheelectromagnetic
mediator (y) is massless, when the weak mediators (w± and zO) are so heavy?
Glashowhad noparticularlygood answer ('Itis a stumblingblockwe mustoverlook',
he said coyly). The solution was providedby Weinberg and Salam, in 1967 (see refs.
[8] and [9]) in the form ofthe 'Higgs mechanism' (Chapter 10). Finally, there a
structural difference between the electromagnetic and weak vertex factors, which
at first glance would seem to preclude any possibility ofunification: the former are
purely vectorial (y'"'
). whereas the latter contain vector and axial vector parts. In
particular, the w±coupling is 'maximally' mixed V-A in character: y'"'(l - y�)
p
n.
of W·s
nol discussed the couplings of W's �nd ZO's One
similar
s
in QeD, and
listed in App"ndix D.
AU
This
s
i
.
• I have
1be ruks ar�
to
to tho e for gluon-gluon coupling
�nother (or
�re
to the hoto )
9. 7 fl«lTO�ak Unificalian
This last difficulty is overcome by the ingenious device of absorbing the matrix
(1 - yS) into the particle spinor itself. Specifically, we define
(9.116)
The subscript (L) stands for 'left·handed', and is supposed to make you think
'helicity -1'. However, this s
i seriously misleading, since UL is not, in general. a
helicity eigenstate. In fact, for solutions to the Dirac equation,
(9.117)
(Problem 9.26). ifthe partick in question is massless, then E = Iplc, and
(9.118)
where
(9.119)
as before. Remember that (ll/l)I is the spin matrix for a Dirac particle, and hence
((:I . I) is the helicity, with eigenvalues ± 1 . Accordingly
1
2
,
-(I - Y )u(P) =
I
0,
u(P),
ifu(P) carries helicity + 1
if u(P) carries helicity - 1
]
(for m = 0)
(9.120)
(If u(P) is lIot a helicity eigenstate, i (1 - y5) functions as a 'projection operator',
picking out the helicity -1 componenl.) On the other hand, if the particle is
lIot massless, it is only in the ultrarelativistic regime (E » mez) that Equation
9.118 holds (approximately), and hence only in this limit that UL (as defined by
Equation 9.116 carries helicity - 1. Nevertheless, everybody calls Ut a 'left·handed'
state, and I shall stick to the customary language:
Meanwhile, for alltipartic/u we definet
(9.121)
• Ple�se unders�nd; Equation 9.116 is the .kfittilWn of "L � nobodys �rguing about Wt. rm
only ....rning
..
you that the """'" is misleading, ·left·handed· does rwI mean 'helicity -1". except
in CQntnts where the partide's m.1SS is negligible.
t Ifthe sign of yj seems strange, refer to the fOOlnote following Equation 7.30.
1 339
340
I 9 Weak !nluactions
Table 9.2 Chirai spinors
Partides
Antiparticles
R and L co=spond 10 hdidty +1
�nd I if 1ft _ I), _nd appn:>.x;'n"tdy
so if E » m2.
-
The corresponding 'right-handed' spinors are
As for the adjoint spinors, since y
S
S
with y!J. (y!J.y _ _ y y!J.),
Ss
i Hermitian (y St
(9.122)
_
y S), and it anticommutes
(9.123)
Similarly
(9.124)
We call these various spinors (summarized in Table 9.2) 'chiral' fermion states
I emphasize that this is nothing but notation; it is u�ful because it allows us
(from the Greek word for 'hand' - same root as 'chiropractor').
to recast the weak and electromagnetic interactions in a form that facilitates their
the w- (as it occurs, say, in inverse beta decay, Example 9.1):
unification. Consider, to begin with, the coupling of an electron and a neutrino to
,
� 7 flUl�Qk UMiJiUllion
111e contribution to J( from this vertex is given by
);, = liy"
(1 - Y'
,
)
,
(9.125)
(here e and v stand for the particle spinors; for a while we need to keep careful
track of the different particle species, and u" u , elc. just gels too cumbersome).
111is quantity is calle<! the weak 'current'; as we shall see, it plays a role analogous
10 the electric current in QED. Now
••
(9.126)
Y"
'0
y"
(!..::..C ) _ (!..±L)
2
-
2
(9.127)
Y"
(!..::..C) _ (!..±L) (!..::..C )
2
2
y"
(9.128)
2
This may not look like much of an improvement, but it enables us to write
Equation 9.125 more neatly, in terms ofthe chiral spinors:
(9.129)
111e weak vertex factor is now purdy vectorial - but it couples left-handed electrons
to left-handed neutrinos. In this sense it is still structurally different from the funda­
menial vertex in QED; however, we can play a similar game there, too. Notice thai
(9.130)
(similady, U -UL + uRI, so the electromagnetic analog can itselfbe written in lerms
of chira] spinors:
(9.131)
(For future purposes, it is best to build in a faclor of -1, 10 accounl for Ihe negative
charge ofthe electron). Observe that the 'cross terms' vanish:
( I + r' )
( I + Y' ) e
( I - Y' ) ( I + Y' )
,
,
'
tLy"eR = t = ty" ,
,
Y" -
(9.132)
1341
'" I
bu'
(9.133)
Equations 9.129 and 9.131 are beginning to look like the stuffof which one might
build a unified theory. It is true that the weak current only couples left·handed
states, whereas the electromagnetic current couples both types, but apart from
that they aTe strikingly similar. So attractive is this formulation that physicists
have come to regard left· and right-handed fermions almost as different particles:
In this view, the factor (1 - yS)/2 in the charged weak vertex characterizes the
participating particles, rather than the interaction itself; the latter is vectorial in all
cases - strong, electromagnetic, and weak alike.
9.7.2
Weak 1505pi" and Hypercharge
In addition to the (negatively charged) weak current
describing the process e- ...... v, + W-. there is also, ofcourse, a positively charged
current
w<l
�
• Tbtte is a danger in carrying this 100 f�r. You
may find yoursrlf wondering, for rumple,
whe�r the left·hmded electton necessarily
has the same m"ss as the right·handed elec·
tron; or, noting that no vecTOr inte=tion can
couple a Ieft·handed p;lrtide TO a right·handed
one (� Equations 9.112 and 9.113), you
may �sk now the two 'worlds' communica�
a! all. Both questions art based on a mis·
understanding of Ul and u�. "The problem
is that, usrful:as it is in describing p;lrtide
iIIumctio/lj, h�ndedness is no! conseMd in the
prop;lgation of a frte p;lrtide (unless il$ =55
is uro). Formally, )'1 does not commu�
with the free·p;lrtide Hamil!onian. In fact,
u, and w. do not samfy the Dirac fiJuation
(� Problem 9.26). II p;lrticle that starts out
left·handed will soon evo]"" a right·handed
component (By contrut, lulkity is conserved
in free·p;lrtkle prop;lgation.) Only for mal.1ess
fennions can ]"ft· and righ!.handed species be
considered distinct p.1rtides n
i the full "'.....,
of the word.
9.7 ElearoweoK Unification
representing the process II. -+ e- + w+ . We can express them both in a more
compact notation by introducing the left·handed doublet
(9.134)
and the 2 x 2 matrices
(9.13S)
so that
The matrices r '"'are linear combinations of the first two Pauli spin matrices
(Equation 4.26):
(9.137)
(I use the letter r here, instead of u, to avoid any possible confusion with ordinary
spin.) This is all very reminiscent of iS05pin: in Se<;tion 4.5, we put the proton and
neutron into a doublet similar to Equation 9.134. Indeed, we could contemplate a
(0 )
full 'weak isospin" symmetry, ifonly there were a third weak current, corresponding
1
0 ,
to ! r 3 = !
l
l
-1
(9.138)
'Perfect!' (I hear you exclaim), 'There's the m:utral weak current!' Not so fast. This
current only couples kfi"handed particles; in the older language, it is pure V·A,
whereas the neutral weak interaction involves right.handed components as well.
But hang on - we're almost there.
hypercharge (Y): which is related to electric charge (Q, in units of t) and the third
Building on the parallel with isospin, we are led to consider a weak analog of
component ofisospin (ll), by the Gell·Mann-Nishijima formula (Equation 4.37):
(9.139)
We introduce, then, the 'weak hypercharge' current
(9.140)
• You h�ve prob.lbly forgotten this word. but hyp.n:karg< i$ �$S�nti.ally the $ame as strangeness.
only shifted, in the GO"" of b.lryons. sO th�t th� c�nter rOW of Eightfold Way di.-.grams wiD al·
w�ys carry Y � I). Specifically. Y � S A. whMe A is th� baryon number.
...
1 3043
1«
I9
Weuk Interoaions
does not touch right-handed components at all. and the combination
This is an invariant construct. as far as weak isospin is concerned. for the latter
is itselfinvariant.· The underlying symmetry group is called SU(2)L 0 U(I); SU(2)L
refers to the weak isospin (with a subscript to remind us that it involves left·handed
states only). and U(I) refers to weak hypercharge (involving both chiralities).
! have developed aU this in terms of the eie.::tron and its neutrino. but it is a
simple matter to extend it to the other leptons and quarks. From the left·handed
doublets (cabibbo-rotated. in the case of the quarks)
(9.141)
we construct three weak isospin currents
(9.142)
and a weak hypercharge current
(9.143)
wherej:;" is theele.::tric current:
,
J:m = L Q;(ii;LYI'UjL + U,RYI'UjR)
(9.144)
••t
{summed over the particles in the doublet. with Q. the electric charge).t
• If you UTe to Ihink of it this w�y. what
� h�ve done is to combine two weak
s
i
(�). Pretty i i
sm lar . . . is there anything
isospin doublets to make an isotriplet.
10 Ihis? Nope. After all. (i) we�1: i",,"pn
i ap'
plies 10 leptons as weD as quarks (and to all
/; the Lost. together with a right·handed pi�e.
m�kes the we�k hypercharge current. /.
involves only the Itft-"�",kd chiralities. (all
"iiLtL. (liLVt - <L�d. <tV, (an:ologom to
Equation 5.38). and an isosinglet (liLVt + 'ted
(analogous to Eqmtion 5.39). � first three
8" to mau the weak isospin currents l'and
t You might ask what Ihe diffi...na is be�n
""''''' isospin (and hypercharge) and their or·
dinary ('strong') counterparts. The question
is JUrticul�rly pertinent when you come 10
the light quarh: the weak isospin doublet is
(;)
t' whereas Ihe strong i"""pin doublet
Ihree qmrk generations); (ii) weok isospin
right.handed states are nngllU - that is. in·
v�riant - �s far as weak siospin is concerned);
jiii) weak isodoublets are Cabibb'Hotated. To
put it pbinly. strong isospin and weak isospin
have nothing to do with one another. s,ave for
a common mathematical structure (which. for
that mailer. the� sh�re with many other sys·
tems. such as ordinary spin t) and the (per.
haps unforrut1llte) simil.o.riry in tlleir n�mes.
9_7 fltclro�Qk Unification
9.7.3
Electroweak
Mixing
Now, the GWS model asserts that the three weak isospin currents couple, with
strength g"" to a weak isotriplet ofvector bosons, W, whereas the weak hypercharge
current couples with strength g /2 to an isosinglet, B:
(9.145)
(Ibese four particles correspond, ultimately, to the weak and electromagnetic me­
diators: W±, ZO, and y - but with a twist, as we shall soon see.) I use bold face
here to denote a three-vector in weak isospin space; the dot product can be written
out explicitly:
i,,'W" =j1 Wl'l +.i! W..2 +j! w,,)
(9.146)
or, in terms ofthe charged currents,j; = j1 ± y;':
(9.147)
where
(9.148)
are the wave functions representing the wi: particles.
The couplings to w± can be read off immediately, from the coefficients of
W; in Equation 9.147 For example, in the process �- -!o v, + W- we have
jj; = vLy..tr = vy.. [(t - yS)/2J�, giving a term
(9.149)
The vertex factor is
(9.150)
which is exactly what we started with (Equation 9.5).
But the two neutral states (Wl and B) 'mix', in Glashow's theory, producing one
massless linear combination (the photon), and an orthogonal massive combination
(the ZO):
AI' = B.. cos 9", + W! sin9",
2" = -B" sin9", + W! cos8",
(9.151)
134$
3461
9 Wwk In'�rQl;1ion�
(You see why 0.., is called the 'weak mixing angle'.) In terms of the physical fields
(A" and Z"), the neutral portion of the electroweak interaction (Equation 9.145)
reads as follows:
-i
[g..,j! W"l + �j!W] -j {[g..,sinO..,j! + � COSO..,j!] A"
+ [g..,COSO..,j! � sin O..,j!] Zil l
=
-
(9.152)
Ofcourse, we know the electromagnetic coupling; in the present language it is
(9.153)
-igd:;'A"
j! �j!. Consistency of the
Meanwhile, from Equation 9.143,):' = +
troweak theory with ordinary QED requires
0..
g sin = tcosO... = g.
...
unified elec­
(9.154)
Evidently the weak and electromagnetic coupling constants are not independent.
There remain the weak couplings to the ZOo Using Equations 9.143, 9.152, and
9.154,we obtain
(9.155)
where
�
..
= sinO... cosO..,
(9.156)
From Equation 9.155 we can pick out the neutral weak couplings. For example,
the process v. -. v. + ZO comes exclusively from the tenn; referring back to
Equation 9.138, we have
j!
(9.157)
and hence the vector and axial ve<:tor couplings (Equation 9.90) are 'v = ,� = ! .
I'll leave it for you to work out the other entries in Table 9.1 (Problem 9.28)'All this raises some obvious questions: by what mechanism is the underlying
SU(2)L ® U(l) symmetry of the electroweak interactions 'broken'? Why do the B
and Wl states 'mix' to form the ZO and the photon? If weak and electromagnetic
interactions are, deep down, both manifestations of a single ele<:troweakforce, how
come the weak mediators (W*and ZO) are so heavy, while the electromagnetic
mediator (y) is massless? I'll address these questions in the next chapter .
• Sinc� th� weak mixing anglr is undetermined n
i th� GWS model, th�r� r�main. in effttt, IIW
indep"ndent <:oupling constants (g. and g... R�, or g. and g.): in this sense, it is not a com·
pletely wnijied theory, but rather an inUgrUtul tMory of �ak and eiectrOIn:rgna;c interactions.
t Cabibbo, N. (19f(3) Physieal ReWw
WIm, 10, 531.
l For mor� d�ailed cakubtions n
i
_ali; inter;>etion theory consult
the classic tr�atise by Marshak,
R. E., Riazuddin and Ry.m, C. P.
in ui, C. H . (ed) (1981) GClup
Theory of W.:ak ClIId Elu:tromag·
lleric 'nuractiom, World Scien·
tific, Singapore. S� abo (b) Wein.
berg's Nobel Prize lecture, reprinted
n
i (1980) Sciena, 210, 1212.
0969) Thwry of WwJ: /IIkllJdiDm ill
to 'I Hooft, G. om) Nuckor Physll:s,
Pa.rfide Physll:s. John Wilt')' &. Sons.
New York; or the briefer account
by (a) Commitu. E. O. (1973) WwJ:
B31, 173: and (19'71) 8)5, 167.
/IIkrltCtw..s, McGraw·Hill, New York.
For a comprehensive review of weak
o
i teractions in th� quark mood, $�
(b) Donoghu�, J. F., Golowkh. E. and
Holstein, B. (l9S<i) PhyJlGs Reports,
131, 319; or (c) Commitu, E. O.
and Budsbaum, P. H. (1983) Wwk
/IIIUl>CIWIIS ofupltm$ 01,..1 Qu<lrh.
Cambridge University Press, Cam·
bridge.
1 Glashow. S. L. lliopoulos, J. and
MaLani, L (1970) PhyJicoll RtWw,
02. 1585. This and the other fun·
dam�ntal polpeTI on weak inler;>e·
C. H. (ed) (1981) GCllI£e Theory of
WwJ: Clm! E1ectromcapaic /llurev
liollS, World Sd�ntilic. Siogapor�.
lion �ry an reprinted in (a) Lai,
• Kobayllshi, M. and Maskawa. K.
(1973) Progress ill Theoretical Physics.
.9, 652.
5 Differ�nt authors us� different p;lI'
nm�teri�tions: J follow th� Re·
IMw of !'CInidl: Ph)"Sic:J convention.
, !'CIrtick Ploysics Booi:kl (2006).
For a useful discussion of the
CKM matrix, see Cheng, T. ·P.
and Li. L ·F. (198.) GGuge The.
ory of E!e="tClry PCll1icu Physics.
Oxford, N�w York. Sect 12.2.
7 Bludman, S. A. (1958) NIWI'O Ci·
�.9, «3.
• Glashow. S. 1. (1961) Nuckor
C. H. (ed) (1981) GCluge' Theory of
Physics. ll, 57<J. Reprinted n
i (al Lai,
WtaS: Clm! Ekaromagnetic '"IUItC·
Reprinted in (a) Lai, C. H. (ed)
(1981) Gauge Theory of Wellk
01,..1 El«U"otn-agIItlic /IIUrlIClions,
World Scientific, Singapore.
11 Hu�rt, F. J. � al. (1973) Physics Let·
un, 45B, 138:
(197.) Nuckar Physics,
B73. I.
11 Meanwhil� a sm<":S of � inelH'
perimenis (also at CERN) not only
tic IlftLtrino-proton scattering B;.
supported th� bo.sk structure or
charged and neutral weak int�rac·
lions but also h�lped to confinn
the quark model ilStlf. See. for
exa.mpl�, Perkins, O. H. (2000) /11'
trod"" ion 10 High E"",!), '*)'SiI:l:,
.th edn. Cambridge University
Press, Cambridge, UK. Sect 8.7:
(a) Halzen, F. and Martin, A. O.
(19...) QUIlrks Cllld UptOllS. John Wi·
ley & Sons, N�w York. Sects. t2.7
and 12.10: {b) Close. F. E. (l9:>g)
An /IIlroductioll 10 QuGrb ClIId !'CI,.
loll$", Academic, london. Sect. 11.3.
U AmUon. G. � al. (1983) Phji$ics
l.tturs, lllB. 103; and (1983)
126B, 398. For a revi�w of these
discoveries, se-e (a) Radermacher,
E. (1985) Progms n
i Panicu
ClIId NUCUClr Physics. 1.4, 2)1.
l' Data on u"
+ .- ..... u. + j.C are
from Vilain, P. �t al. (199S) Ph)'$ics
Letun B. 36-4, 121; data on u" + ,­
v. + e- are from (a) Ahrens,
1. A. � al. (1990) Ph)lJical Rwitw D.
_
P. et al. (1978) Ph)'$icJ Letkn, 7.8,
41, )297. Earlier data by (b) Alibr.m.
422: which _re inconsistent with
tfullS, World Scientific, Singapore.
9 Weinberg. S. (1967) Physi.:al Re·
vitw LeIUrs, 19. 12(,.4: (a) Salam.
the GWS mod�l. turned out to be
ory, (eds N. Snrtholm), Almquist
corrected by (c) Armenise. N. � at
A. (1968) E!e_IIUlry PClrride The·
and wiksdl. Siockhoim. reprinttd
consternation at the tim�: they were
wrong. although they caused som�
(1m) Ph)'Sil:l: LeI/US, 86B, 225.
141
I 9 Wtok IlIItractions
15 Fortson, E. N- �nd Wilets, L (1980)
Advances in Atomic and Moltc,,lar Ph)'l"ics, 16, 319; (a) Pres(o",
C. Y. et al. (1978) Pkysicl /.titers.
77B, 347; and (1979) 84B, 524:
(b) Wu, S. L (1984) Pkysica1 Re·
POrlS, 107, 229; (c) Wood, C. S.
d aI. (1997) Sdtnu, 275, 1759.
16 See, for ex;omple. Frauenfdder. H.
and Henley, E. M. (1991) Subawmic
Pky'ics, 2nd Mn, ?rentice-Hall,
Englewood Cliffs, N.J. Sect. 5.7.
I' Even bolder in 1957. when
Schwinger laid the �sential ground·
work for such a theory: Schwinger,
j. (19S7) Annals of Pk)'l"ics {New
Yori:}, 2, 407. Reprinted in (a) 1.;o.i,
C. H. (ed) (1981) Gouge Theory of
Weak o,.d EfulrOlII<lg1i1et c /,.ltTOC­
liollS. World Scientific, Sing�pore.
Problems
9.1 Derive the completeness relation for a massive particle ofspin 1 (see Problem 9.27 for
the mnsless �n;llog), (Hint: �t the z uis point along p. First construd three mutually
I»
orthogonal polarization vectors {{ ll, { 2I, { ) that �tisfyp"{u = 0 and {I'll' = -1.(
i � �
(9.158)
9.2 Calculate the trace
for arbitnory (real) numbers Cv and c".
[Answer : 4(et + c!)[Pjpi + Pl� - PI
. P2 8"OJ
(9.159)
9.3 (0) Calculate ( 1.A12 ) for v" + t- ..... jA.- + v, using the more general coupling )'''
(1 + f )'1). Check tlut your a nsW!'r reduc� to Equation 9.11 in the case ( _ -1.
(hI Let m. - m" - 0, and calculate the CM differential scattering cross section. Also, find
the IoUl/ cross section.
(el /fyou had accurate experimental data on this reaction, how could you determine {�
9.4 Show that Equ�tion 9.30 is equivalent to EqUlltion 9.29.
9.S By making the appropriate chang� in Equ.:ttion 9.35, determine the lifetime of the r
lepton, pretending the deay is purely leptonic. (Assume also that the muon mass can
be neglected. in comp<lrison with m,_) Comp<lre the experimental value.
9.6 Suppose the weak n
i teraction WBe pun: \'tela. (no y} in Equation 9.5). Would you still
get the same shape for the gra ph in Figure 9_1�
9, 7 Eleclr�Qk Ul1ificaliol1
9.7 What is the Quen:l8l' vilue ofthe electron energy in muon decay?
9.8 Using the coupling ),"(1 + � )'1) for n ...... p + W, but ),"{l - )'1) for1M leptons, cakuIate
the spin·ave"'ged amplitude for neutron bet:I decay. Show ih;ot your result reduces to
Equation 9.<41 when f �-\.
+(p, . P4}!P2 . Pl}(l + (i)l
-
]
(I - (il)mpm"c1 (pz . P4}1
9.9 (a) Derive Equation 9.52. (bl Derive Equation 9.58.
9.10 In the !ext. I said that electron energies in neutron decay range up to about (m" - m,)c2.
This is not =1, since it ignores the kinetic energy of the proton and 1M �lltrino.
What kinematic configur.ltion gives the m;u:imum electron energy? Apply conservation
ofenergy and momentum to determine the oacl milXimum electron energy.
How far offis the approxinute anS_r (give the percent error)?
9.B la) Integrate Equation 9.59 to get Equation 9.&0.
(bJApproximate as suitable for m.« t:. m � (m� - m,). Note that m. now drops out.
9.12 Obtain Equation 9.62.
9.\3 Find the minimum de Broglie wavelength (l. • "!p) of 1M W in neutron decay.
and compare it with the diameter of the neutron (� 1O-1l em). [Amwt:r: maximum
Ipl
=
1.18 MeV/c, occurring when p and t emerge back to back. so the minimum A �
to-LOem]
9.H Anilyze ,.,- decay as a scattering process, using themethods of Example 7.8 and Section
8.5. Calculate the decay rate. and. by comparing your answer with the one in the text,
obtain the fonnula forf. n
i tenns of(+(O)l2. Take the quarks to be massless.
U5 Show that f
i mel « E
where II is a particle spinor satisfYing 1M Dj",c equation:
(Eqs. 7.35 and 7.41). Show therefore Uut the proje<:tion ""'In>:
13<49
""
I
picks OUI th� helicity ± 1 component of u:
9 Weak !I1IUQClions
'.1� Cakul;tte the ratio ofthe decay rates K- ..... e- + v, and Kobserved branching ratios.
Hi Calculate decay n� for the following processes: (a)
O
->
Jl.- + ;;;�. Compare the
EO -> I:+ + e + ii•. (b) E- ....
A + t + v,,{cl S- ..... S + t + ii.,(d) A ..... p + t + ii.. (e) E- ..... l1 + t + ;;,: If) SO .....
1:+ + e + 'ii._ Assume the coupling is always y"(l - yS) - that is, ignore the strong
9.l8 (a) Show that as long as the CKM matrix is .."itw}'· (V-1
interaction corrections 10 the axial coupling - but do not forget the Cabibbo factor,
Compare the experimental d;tta.
_
Vt), the GIM mechanism
- works for thr� (or any number of) genentions. [Note:
.. ..... d + W" carries a CKM factor V,.j; d ..... II + W- carries a fador V:,..
for eliminating � ..... Jl.+Jl.
(bl How mmy independent realparametersare there in thege�nJ 1 x 3 unitary matrixl
How about n x /1) [Hint II helps to know that my unitary matrix (U) can � written
n
i the form U = iH, wh� H is a hermitian matrix. So an equivalent question is,
how rmony independent real p;orameters are there in the general I\t:rmiliall matrix))
We are f� to change the phase of each quark MIlle function (normalization of u
really only determines INll; s� Problem 7.3), $0 211 ofthese p;orameters are arbitrary
- or rather, (In - I), since changing the phase of aU quark wave functions by the
same amount has no effect on V. QlImioll: Can we thus reduce the CKM matrix to a
real matrix (ifit s
i real and unitary, then it is orlhogonal: V-I =
V).
(e) How many independent real p;orameters are there in the genen.! 3 x 3 (real)
orthogonal matrix? How about n x II?
(d) So, what is the answer? Can you re-duce the CKM matrix to real fonn? How about for
only two gen�rations {II �
2)1
9.19 Show that th� CKM matrix (Equation 9.87) is unitary for any (real) numbers OIl, OJ),
en, and d.
9.20 Using th� e�perim�ntal values ofthe F�nni constant Gf (Equation 9.38) and the weak
mixing angle 8. (Equation 9.93), 'prNict' th� rmoss of th� W±and th� z!>, n
i GWS
th�ry. Comp;or� th� experimental values.
9.21 In Exampl� 9.4 I used muon neutrinos, rather than tkctroll neutrinos. As a matter of
bet. Vp and ii.. beams are �asi�r to produc� than v, andii,. but ther� is also a �retical
r�ason why v.. + �- ..... v� + r is simpl�r than v, + e- ..... v, + r Or ii, + e- -10 ii, + e-.
i the GWS
9.2.2 (.) Calculate the differential and total cross section for ii.. + e- -> ii.. + e- n
Explain.
m�l. [Allswer: 5am� as Equation 9.100. only withthe sign of 'A'V reversN.[
(b) Find the ratio u(ii.. + r ..... ii.. + e- )Ju{v� + r ..... v� + r). Assume the energy is
high enough that you can set .... . O.
f +j. wheref is any quark or any lepton. Assume
f is so light {compared to the Z} that its mass can be n�ected. IYou'U n� th�
9.2.3 (al Calculate the decay rate for z!> -10
completeness relation for th� z!> - � Problem 9.1.)
(b) Assuming these are th� dominant decay modes. find th� branching ratio for �ach
species ofquark and lepton (remember that the quarks come in thr� colors). Should
you includ� the top quark among th� allowed decays? (AII"""r: 3% each for e, Il. r :
7% each for v,. v..' v,; 12% each for u. c; 15% each for d . s, P.)
• For experimental confirmation see Problem 9.13.
9. 7 E!eclrOlWok Ullificatioll
(e) Caleul�t'" th", lifctim", ofth", z!J. Quanlil�tivdy, how would it dlang'" ifther'" """r", a
founh genen.lion (quarks and leptons)? (Notie", WI an accun� m"'a5ur",m",nl ofthe
z.O lif�me tells us how many quarks and I",ptons there Gin � with masses I",ss wn
45 GI'Nlc1 .)
9.24 Estimat", R (th'" total l1ltio of quark pair production 10 muon pair production in e+e­
sGittering). when the process is mooiatoo by z.O. For the sake ofargument. pre�nd the
top quark is light enough so that Equation
9.109 un be used. Don·t forget color.
9.25 Graph the ratio in Equation 9.113 asa function of:li: "" 2E/Mze1. u� rz
(Mze1/fi) (Problem 9.23).
=
7.3�/48Jr)
9 2(; (0) If "(P) satisfi",s th", (momentum spae",) Dir3e equation (Equation 7.49), showtlut
.
Ut
and "R (Table 9.2) do lIat (unless III _
0).
(b) Find the eig",nvalues and eigt'nspinors ofthe matrices P:I: - t(1 ± )'�).
(e) Can th"'re exist spinors thaI are simuluneously eigenstates of P (say) and of the
+
Dinc operator r.; - "")?
(Answer : No; these oper.ltors do not commute.]
9.27 Work oul the """ak isospin currents)! and}! for the lighl quark doublel U and rf.
Also, construdthe electromagn�c <:um!nt (f.."') and the weak hypercharge current
(Leav", your answers in �rms of If.)
U:).
9.28 From Equation 9.155. determine the vector and ui;!l vector couplings in Table 9.1.
9.29 In Problem 9.5 you found thedKaynte r for t ...... e + �, + ii" andfor r ..... IL + �, + Vu
(which is essentially the same). Howabout for the hadronic modes (T ...... d + v, + Ii and
T ...... S + �t + II)? Estima� the Iifetim'" of the T (including both leptonie and Iudronic
modes) and the bnnching ratios for the dedron, muon, and Iudron modes. Compare
the experimental values. (PartiQ! allsw.:r: r"" - 5 r]
9.30 (al Estima� the lif�m'" ofth", charmoo quark. (First dec:ide whatmodesdomina�. and
then make the appropriate modifications in the muon dKay fonnw... Equation 9.35)
(b) On the basis of (a). estimate the lif�me of the 0 meson (0" _ eli and 0+ - cd),
(Hilll: Refer to Problem (9.29)1
treating the light quark as a spe<:tator. Also ",stimate the branching ratios for the
various �mileptonit modes and for the hadronic mode. Compare the experimental
values.
(e) In th", same way, estimale the lif�me of the B meson (8" .. bd and B- .. bU). Note
that mor", decay modes ar'" avai
l able 10 the b quark. Find
the branching ratios, and
compare the experimental values.
(d) According to Equation 9.35. the decay rate goes like thefi}th power of th� mass. The
bottom quark is almost four times as massive as the charmed quark. Why. then.
isn·t the lif�me of the D meson
1000 tim�s longer than that ofth", B? InJact. their
lifetimes are quile comparable, but this is som�thing ofa coincidence. Explain.
9.31 Calculate the lifetime ofthe top quark. Note that bec�use III, ". mb + IIIW. th", top c�n
decay into a rlilll W (I
..... b + W+). wh�r�as aU oth",r quarks must go via a virtual W.
As a consequence. its lifetime is much shorter, and wt"s why il does not fonn bound
states ('truthful· mesons and baryons). Take the b quark to � massless (comparoo to '
and
W). (Allswor: 4 " lO-lS s)
9.32 Th� radicai new (your nam",) theory ofweak interactions as�rts that the W actually has
spin
0 (not I). and the coupling s
i ·scalar/pseudo-scalar', inst�ad of·vector/axial.vector'.
Specifically. in your theory th'" W propagalor is
1351
lS21 9
Weak !nleroctians
(replacing Equation 9.4), and th� ven�x factor is
-ig... (l -
2.jj
s
y )
(replacing Equation 9.5). Consider 'inverse muon decay' (v� + t ..... � + v.), in this
theory:
lal Draw the Feynman diagram, and construct the amplirude, A,
(hI Determin� th� spin-averaged quantity, ( [All).
Ie) Find the differential sc.;ottering cross section, in the CM frame, in tenns ofth� dectron
�nergy E and th� scattering angl� (J. Assume E » m� r? » m,e2, so you c.;on safely
neglect the masses oCboth the �lectron and the muon (and, ofcours�, the neutrinos).
(d) Calculate the IOtil cross section, und�r the same conditions.
I�J Bycomparing th� orthodox predictions for this proc�ss, instruct the aperim�ntilists
how best to confinn your theory (and demolish the Standard Model). (Note: lhere
is no reason to sup� that the weak coupling constant (g,.) in your theory has th�
same value as it does in the Standard Model. so a test that depends on this number
is not going to be very persuasiv�.)
'.33 lhe rows (and columns) of a unitory matrix are orthonormal. 1ms suggests a number
of tests of the CKM model, as the values of the matrix elements are measured with
increasing precision. For ex.ompl�, orthogonality of the first and third columns impli�s
(Equation 9.86)
Or (dividing by the middl� term)
1 +Zl +Z2 = 0,
where
Plotted in th� compla plane, the numbers I, ZI, and Z2 must add up to form a dosed
loop, called the 'unitority triangl�'. Look up the best current values for the CKM matrix
elements, and plot I, ZI, and :2. Does their swn in fact form a closed triangle?
9.14 Find the threshold v� �n�rgy forinv� muon decay (Example 9.1), assuming thetirget
electron is at rest. Why is the answer so huge, when all we're doing is producing a
muon?
10
Gauge Theories
This chapter introduces the 'gauge fhwms' that describe aU dmu:ntary partick interac­
tions. I begin with the wgrangian formulation of classical mechanics, and procr�d to
Lagrangianfield fhwty, tht priru:iple oflocal gauge invarianu, the notion oJspontaneous
symmetry·breaking, and the Higgs mechanism (which accountsfor the mass oftk W's
and the Z). This material is quite abstract (in contrast to previous chapters); it COltuntS
thefondarnmtal quantumjitld thwrnsfrom which the Fq'nman rules derive. It will not
htlp you to calculate any cross sections or lifetilm5. On the other hand, the ideas discussed
hert constitute thefoundation on which virtuaUy all motkm theories an: predicataf. To
understand this chapter it wi!! hdp to havtstuditd some Lagrangian mechanics, but more
essential is the relativistic notation in Chapter 3, tht taste ofgroup thwry in Chapter 4,
the Feynrnan calculusfrom Chapter 6, and tht Dirac equatioltfrom Chapter 7.
10.1
Lagrangian Formulation ofClassial Particle Mechanics
According to Newton's set:ond law of motion, a particle ofmass m, subjet:ted to a
force F, undergoes an acceleration a given by
F "nna
(10.1)
If the force is C(ln5f)vatiw:, it can be expressed as the gradient of a scalar potential
energy function U:
F = -VU
(10.2)
and Newton's law reads
d.
m- = -VU
d.
where v is the velOCity [IJ.
I..troduaio.. '0 Ekmen'4ry P4rfidn, Swmd Edition. David Griffiths
Copyright (I 2008 W1LEY·VCH V�rbg GmbH & Co. KGlA. W�;nh";m
ISBN: 973-)·527-40601·2
(10.3)
lS.! IO
CQug� Thtori�1
An alternative formulation ofclassical mechanics begins with the 'Lagrangian'
L = T- U
(10.4)
where T is the kinetic energy ofthe particle:
(10.5)
The Lagrangian is a function ofthe coordinates q; (say, ql = x, '12 = }', ql = z) and
their time derivatives ili (ill
= v�, ilz =
Vf,
ill
= vz). In the Lagrangian formulation,
the fundamental law ofmotion is the Euler-Lagrange equation [2]:
� G�) = :�
(i = I,2,3)
(10.6)
Thus in cartesian coordinates we have
at
aill
at
aq,
�
aT
= mv"
a"
-
au
ax
(10.7)
(10.8)
and the Euler-Lagrange equation (for i = 1) reproduces the x component of
Newton's law, in the form of Equation 10.3. The Lagrangian formulation is thus
equivalent to Newton's (at least, for conservative systems), but it has certain theo­
retical advantages, as we shall see in the following sections (see also Problem 10.1).
10.2
Lagrangians in Relativistic Field Theory
Aparticle, by its nature, is a /ccalized entity: in classical particle mechanics we are typ­
ically interested in calculating its position as a function of time: X(I), }'(I), z(t). Afidd,
calculate one or more functions of position and time: <p;(x, }', z, I). The field variables
on the other hand, occupies some ngion of space; in field theory our concern is to
<Pi might be, for example, the temperature at each point in a room, or the electric po­
introduced a Lagrangian L that was a function of the coordinates, q;, and their time
tential V, or the three components ofthe magnetic field B. In particle mechanics, we
density) .¥, which is a function ofthe fields <p; and their x, }', z, and t derivatives:
derivatives, il;: in field theory we start with a Lagrangian (technically, a Lagrangian
(10.9)
In the former case, the left side of the Euler-Lagrange Equation 10.6 involves only
the time derivative; a rdatiuistie theory must treat space and time coordinates on
10.2 Lagrongia"s in Rtlali�l$lic Field Theory
an equal footing, and the Euler-Lagrange equations generalize in the simplest
possible way, to:
iJ" iJ(iJ"ofJ;)
( 'It' ) = ,It'
Example JO. J
iJofJi
(i = 1,2,3, . . .)
have a single, scalar field variable ofJ, and the Lagrangian is
The Klein-Gordon Lagrangian jor a Scalar (Spin·O) fidd
1
1
.z = -(iJ ofJ)(iJ"ofJ) _ _
1 "
1
(�)'
,
_
ofJ2
(10.10)
Suppose we
(10.11)
In this case,
(10.12)
(If this confuses you, write out the Lagrangian 'longhand':
In this form, it is clear that
and so on.) Meanwhile
and hence the Euler-Lagrange formula leads to
{IO.OJ
which is the Klein-Gordon equation (Equation 7.9), describing (in quantum field
theory) a particle of spin 0 and mass m.
spinor field of! , and the Lagrangian
Example 10.2
_
The Dirac Lagrangian jor a SpinOf (Spin-!) Field Consider now a
1 355
JS61
10 Goug� Thwries
We treat 1/1 and the adjoint spinor ifi as independent field variables." Applying the
Euler-Lagrange equation to ifi, I find
,:"
"" 0,
03(03l'ifi)
so that
iyl'031'1/I -
c;;) '"
(10.lS)
"" 0
This is the Dirac equation (Equation 7.20), describing (in qwntum field theory) a
to if!, we obtain
particle ofspin ! andmass m. Meanwhile, ifwe apply the Euler-lagrange equation
and hence
which is the adjoint of the Dirac equation (see Problem 7.1S).
l1li
Example 10.3 The Pr()(;a Lagrangianfor a Vectcr (Spin. J) Field Finally, suppose we
take a vector field, AI' , with the lagrangian
(10.16)
Here
az "" -1 " " 03· "
---(a A - A )
a(03I'Ao)
(10.17)
411"
(see Problem 10.1) and
,:"
03A.
1
==
411"
(�)' A,
Ii
(10.18)
so the Euler-lagrange equation yields
(10.19)
rually dg/ll inde�dent fi�lds her� Ii runs
linear combinations of these eight will do
just as _D. and w<: chOOSt to US<: the four
�ach of the four components of t. BUI in
oft·
• Since 'It is a compl�x spinor. th�I� are ac·
from 1 10 8): the real and imaginary p;orts of
applying the Euler-Lagrange �uations any
components of "iF plus the four components
10.2 lQgrongiolll in RelotMslic Fjeld Theory
This is called the Proca equation; it describes a particle of spin 1 and mass m.
Incidentally, since the combination (a"A" - a" A") occurs repeatedly in this theory,
it is useful to introduce the shorthand
(10.20)
Then, the Lagrangian reads
(10.21)
andthe field equation becomes
(10.22)
If the notation is beginning to remind you of electrodynamics, it's no acci·
dent, for the electromagnetic field is precisely a massless vector field; if you
set m = 0 in Equation 10.22 you're left with Maxwell's equations for empty
space: _
The Lagrangians in these examples came out of thin air (or rather, they were
i such a way as to reproduce the desired field equations). In classical
concocted n
particle mechanics, L s
i dtrivtd (L T - U), but in relativistic field theory it' is
usually taken as axiomatic - you have to start somewhere. The Lagrangian for a
particular system is not unique; you can always multiply it' by a constant, or add a
constant - or for that matter the divergence ofan arbitrary vector function (o"M",
where MI' is any function ofI/J; and OI'I/J;); such terms cancel out when you apply the
Euler-Lagrangeequations, so theydo not affectthe field equations. In this sense, the
factors of in the Klein-Gordon Lagrangian, forexample, are purelyconventionaLt
Apart from that, however, what we have here are the Lagrangians for spin O. spin
and spin 1. So far, however, we are talking only offru fields, with no sources or
interactions.
=
l
},
• Notic� tint in this fonnulation A." is th� fun·
d;om�n"'l q""ntity �nd F�' is just conveni�nt
no"'tion (Eq""tion 10.20) - th� reverse of tll�
pfisp«tive "'ken n
i classical d�lIodynamics.
wh�r� E and B (Il�nc� F�') �re fundamen.
",1 and � poteniols
t
are constructs. In po.r·
ticular, for PUIJ'OS"s of the Euler-Lagrange
e<[uations th� 'fi�lds' ar� the components of
A�. '101 F�·.
t The lagrangion (I) carries units of energy
(Eq""tion 10.4), and the lagrangian d<MIity
(.2') has the units ofenergy p<r ..M;I 1'01.."",.
Thefo:Ids carry dim�nsions as foUows:
I/J (scalar field): ./Mi./T
1/1 (spinor field): L-l/1
AI' (vector field): ./Mi./T
These are chosen so that 'it will go o""r to the
Schrodingcr wa"., function (in the nonrela·
tivi"tic limit) �nd A� to the M�ll vector
potential (in the nonquantum limit). By the
way, in Heaviside-l.orentz units the Proca
and Maxwell Lagrangians are conoentionally
multipUed by 4"..
1357
358 1
10 Gouge Theories
Example lOA
The Ma)I'Well Lagrangian for a Massless Vector Field with SOlJfU )/'
Suppose
(IO.2l)
where Fl'v (again) stands fOf {(lI'Av - (l" AI') andjl' is some specified function. The
Euler-Lagrange equations yield
o"FI'V =
4J!"
)"
,
(IO.24)
which (as we found in Section 7.4) s
i the tensor form of Maxwell's equations,
describing the electromagnetic fields produced by a current ]1'. Incidentally, it
follows from Equation 10.24 that
(lV)" = 0
(10.25)
That is, the internal consistency of the Maxwell Lagrangian (Equation 10.23)
requires that the current satisfy the continuity equation (Equation 7.74); you
can't just put in an}' old function for ]I' - it's got to respect conselVation of
charge.
_
10.3
Loa! Gauge Invariance
Notice that the Dirac Lagrangian
is invariant under the transformation
(global phase transfonnation)
(10.26)
(where 0 is any real number), for then � -+ e-i8�, and in the combination �l/t
the exponential factors cancel out. (Already in nonrelativistic quantum mechanics,
ofcourse, the overall phase ofthe wave function is arbitrary.) But what if the phase
factor is different at different space-time points; that is, what if B is a jUnction
ofxl';
(local phase transformation)
(10.27)
Is the Lagrangian n
i variant under such a 'local' phase transformation? The answer
is no, for now we pick up an extra term from the derivative of0:
(10.28)
so that
(10.29)
For what follows, it is convenient to pull a factor of -(qjfic) out ofB, letting
(10.30)
where q is the charge of the particle involved. In terms ofA, then,
(10.31)
under the local phase transformation
(10.32)
So far, there is nothing particularly new or deep about this. The crucial point
comes when we dtmand that tlu compJeu Lagrangian b� invariant undtr local phase
transformations." Since the fra Dirac Lagrangian (Equation 10.14) is not locally
phase invariant, we are obliged to add something, in order to soak up the extra
term in Equation 10.31. Specifically, suppose
(10.33)
i some ntw field, which changes (in coordination with the local phase
where AI' s
transformation of >/t) according to the rule
(10.34)
AI' - AI' +<11').
This 'new, improved' Lagrangian s
i now locally invariant - the <1,,A in
Equation 10.34 exactly compensates for the 'extra' term in Equation 10.31. The
price we have to pay is the introduction of a new vector field that couples to 1/t
through the last term in Equation 10.33 (see Problem 10.6). But Equation 10.33
isn't the whole story; thefoU Lagrangian must include a 'free' term for the field AI'
itself. Since it's a vector, we look to the Proca Lagrangian (Equation 10.21)
-1
1
2' � --FI'"FIL + 161T
" 81T
(-m,, ) , A"A
"
But there is a problem here, for whereas FI'" := (ijl'A" - a"AIL) is invariant under
Equation 10.34 (as you should chec.k for yourself), A"A" is not. EvidtnUy th� new
fi�1d must be massless (m,.., 0), othelWise the invariance will be lost.
n
"'"
• I know of no compdling physical argum�nt
for insisting that a global invarian"" lkould
hold loc.ally. If you belie"" that pha.., trans·
fornutions ar� in some s�n'" 'fund.1mental',
then I suppose one should ]", able to) (any
them ()Ut ind�dently at sp<I""Hke 5ep<oTated
points (which art. after aU. ()Ut of commu·
nication with one another). But I think this
],.,gs the qutStion. Better, for the moment at
kast, to take the rt<ju;",ment oflocal phasr
invarianct as a n�w principk of physics in its
own right.
360
I 10
Gouge Theories
Conclu.sion: If we start with the Dirac lagrangian, and demand local phase
invariance, we are forced to introduce a massless vector field (A"), and the
complete Lagrangian becomes
(10.3S)
As you will have guessed, A" is nothing but the electromagnetic potential; the
transfonnation rule for AU (Equation 10.34) is precisely the gau� invariance' we
found back in Chapter 7 (Equation 7.81), and the last two terms in Equation to.3S
reproduce the Maxwell Lagrangian (Equation 10.23), with the current density
(10.36)
Thus the requirement of local phase invariance, applied to the free Dirac la·
grangian, generates all of electrodynamics and specifies the current produced by
Dirac particles.
This is a truly breathtaking accomplishment. The critical step was the added
term in Equation 10.33. How was this obtained? The difference between global
and local phase transformations arises when we calculate derivatives of the fields
(Equation 10.28):
a.. 1Jr ->- e-iqJ./fI£ a" - i
[
;t-(a,,)..)] 1Jr
(10.37)
Instead of a simple phase factor, we pick up an extra piece involving a..).. . Ifin tht
original (free) wgrongian IVt' nplace eVf'1)' derivative (a..) by the so<alled 'covariant
derivative'
!D.. ii a.. + i
tA..
(10.38)
(and every a" by !D") the gauge transformation of A.. (Equation 10.34) will cancel
the offending term in Equation 10.37
(10.39)
and the invariance of Z is restored. The substitution of !D.. for a.., then, is
a beautifully simple device for converting a globaUy invariant lagrangian into
a locally n
i variant one; we call it the 'minimal coupling /"Uk'.t But the covariant
• Beuusr of thr connrction with gau� n
i vari­
an", in mssical rlectrodymmics. � nOW call
Eqmtions 10.34 and 10.26 'gauge transforITlil'
tions', A� is called tM 'gauge field', and thr
trick in dassicil electrodynamics for obtainin8
the equation of motion for a charged pani.
ca
cle n
i the presence of electrodynamic fields. It
amounts, in this �nsr, to a soph isti ted for·
entire stntegy is caned 'gauge theory.
mu�tion of the Lorentz fot'" law, [n modern
than the principle ofloc.al gauge in""rian",.
invman", as fundamen�land minimal cou·
t The minima! couplin8 rule is much older
In terms of momentum (p� .... ilia.) it reads
p• ..... p. - i(q/c)A., and i. a well·known
I"'rticle throry � p ef�r to "'gard local gauge
l
piing as the vehicle for achieving it.
lOA YQllg-MjJJs Th�0'l'
derivative introduces a newvector field (A,,), which requires its ownftu Lagrangian;
if the latter is not to spoil local gauge invariance, we must take the gauge
field to be massless. This leads to the final expression (Equation 10.35), which
people in the know would immediately recognize as the Lagrangian for quantum
electrodynamics - Dirac fields (electrons and positrons) interacting with Maxwell
fields (photons).
The idea of local gauge invariance goes back to Hermann Weyl in 1918 [3J.
However, its power and generality were not fully appreciated until the early 19705.
OUf starting point (the global phase transformation in Equation 10.26) may be
thought of as multiplication of'" by a unitary 1 x 1 matrix:
"' _ U1/I,
where
Ut U = 1
(10.40)
(here, U = till) . The group of ell! such matrices is UtI) (s� Table 4.2), and
hence the symmetry n
i volved is called 'UtI) gauge invariance'. This terminology
is e1Clravagant for the case at hand (a 1 x 1 matrix is a number, so why not leave
it at that?). but in 1954 Yang and Mills (41 applied the same strategy (insisting
that a global invariance hold locally) to the group SU(2), and later on the idea was
extended to color SU(3), producing chromodynamics. In the Standard Model. aU
ofthe fundamental interactions are generated in this way.
10.4
Yang-Mills Theory
Suppose now that we have two spin. ! fields. 1/1 1 and 1/11. The Lagrangian, in the
absence ofany interactions, is
(10.41)
It's just the sum ofthe two Dirac Lagrangians. (Apply the Euler-Lagrange equations
to this Z, and you'll find that both 1/11 and 1/12 obey the Dirac equation, with the
appropriate mass.) But we can write Equation 10.41 more compactly by combining
"' 1 and 1/12 into a two·component column vector:
(10.42)
(Of course, 1/11 and "'2 are themselves four·component Dirac spinors, and you
might prefer a double-index notation: "'a.i, where a = 1, 2 identifies the parncle
and i = I, 2, 3, 4 labels the spinor component. However, in the present context we
are only concerned with the particle index, although the Dirac matrices, of course,
act on the spinor indices.) The adjoint spinor is
(10.43)
1361
362 1 10 Gouge TMories
and the Lagrangian becomes
(to.44)
where
(10.45)
is the 'mass matrix'. In particular, if the two masses happen to be equal
Equation 10.44 reduces to
(10.46)
This looks just like the one·particle Dirac Lagrangian. However, 1/1 is now a
two·element column vector, and .!i! admits a more general global invariance than
before:
(10.47)
where U is any 2 x 2 unitary matrix
(10.48)
For under the transformation in Equation 10.47,
(10.49)
and hence the combination �1/1 is invariant. Now, just as any complex number of
modulus 1 can be written in the form tf8, with real 0, so any unitary matrix can be
written in the form [5]
(to.sO)
i Hennitian (Ht "" H).' Moreover, the most general Hermitian 2 x
where H s
2 matrix can be expressed in terms of four rea! numbers, al, al, al. and 8
(Problem 10.10):
H ",, 8 1 + t · a
•
In INlrix l!leo!)' the n�lur�l generaliz�tion of
complex conjugation (*) is Hermiti.1n conju·
gation It) - Iranspose conjugation. Of cours.e.
thtre's no distinction in the c.ase of 1 " 1 ma·
trices (complex num�s). but for higher di·
mensions it is the Hennitian conjugate Wt
(10.51)
shares the most useful properties of ordinary
complex conjugation. In this sense the dos·
est analog to a real number (ii '" a*) is a Hu·
mil;"n matrix IA '" AI), and the analog to a
number of modulus I (ii*ii '" \) ;s a un'lary
nullix (ALA _ 1).
10.4 Yang-Mills Tht01Y
where 1 is the 2 x 2 unit matrix, fl, fl, Tl are the Pauli matrices (Equation 4.26),
and the dot product is a convenient shorthand for flal + f2a2 + fl ill. Thus any
unitary 2 x 2 matrix can be expressed as a product:
(10.52)
We have already explored the implications of pilau transformations (t"6); in this
section we shall concentrate on transformations of the form
(global SU(2) transformation)
(10.53)
The matrix dr.• has determinant 1 (see Problem 4.22), and therefore belongs to
the group SU(2). Generalizing the terminology of Section 10.3, we say that the
Lagrangian is n
i variant under global SU(2) gauge transformations: What Yang
and Mills did was to promote this global invariance to the status ofa local invariance.
The inspiration and the strategy were similar to Weyl's, but the implementation
is more subtle; in fact, it's quite remarkable that it works at all. The first step is
to let the parameters (a) be functions of XIL (as before, !'lliet .l..(x) i: -(fic/q}a(x),
where q is a coupling constant analogous to ele<:tric charge):
>/! -+ S>/!, where 5 := £-i<!r·�j%)/� (local SU(2) transformation)
(10. 54)
As it stands, !f is not invariant under such a transformation, for the derivative
picks up an extra term:
(10.55)
The remedy, again, is to replace the derivative in !f by a 'covariant derivative',
modeled on Equation 10.38, but taking into account the structure ofEquation 10.55:
fit,, !!!! a"
+ itT · A"
(10.56)
and assign to the gauge fields AIL (it takes thru ofthem this time) a transformation
rule such that
(10.57)
for then the Llgrangian (Equation 10.46) will clearly be invariant.
[t is not a trivial matter to deduce the transformation rule for A" from
Equation 10.57 (61. I'U leave it for you to show (problem 10.11) that A" -+ A�,
where A� is given by
(10.58)
• Ii is also invariant und�r th� largor group U(2). But Equation 10.S2 shows th..t any el�ment
of U(2] can be expressed ;lS ..n element of SU(2) tim"l; ..n appropriate pha� faclor (in
group"theo"'tical langua�, U(2) = U(I) ® SU(2U, and since we have al",ady studied U(I)
..Ii..nce. the only thing /lew here is the SU(2) symmetry.
;nv
1 363
364
I 10
GOI/go T�oril!S
This much is relatively straightforward. But 5 and 5-1 in the first term cannot be
brought together. because they do not commute with r . A". Nor is the gradient
of 5 simply -i(qr . iJ"l./lic)5. because 5 does not commute with r . iJ"l.. You can
work out the exact result (using Problems 4.20 and 4.21). if you have the energy.
but the answer is not particularly illuminating. For our purposes it will suffice to
know the approximate transformation rule. in the limiting case of very small )l.I.
for which we may expand 5 and keep only the first·order terms:
5 :;::;' 1 - � r . l..
5-1 ;::;: 1 + �r l..
0,,5 :;::;' -
:t .
iJ"l.
(10.59)
In this approximation Equation 10.58 yields
-r · A� ;::;: r . A,, + �[r . A". r . l.J + -r . iJ"l.
(10.60)
and hence (using Problem 4.20. to evaluate the commutator)
2,
(l. x A,,)
Iic
�
AI" = A" + iJ"l. +
(10.61)
The resulting Lagrangian
"
.!t' = ilicty :1),,1/' - me2t1/' = [ilicty"iJ,,1/' - me2t1/'J - (qty"-r1/') · A"
(10.62)
is invariant under local gauge transformations (Equations 10.54 and 10.58). but we
have been oblige<! to introduce three new vector fields A" = (Aj. A�. Ai). and they
will require their own fru Lagrangian:
I p;' F
I p;' F
I .
2'/1. = = - 11" F" . F".
F 1611" I ,,'1 1611" 2 ,,'2 - 1611" 1 "'1
16
(10.63)
I
p,"
(Again. the three·vector notation pertains to the particle indices.) The Proca mass
term
1
(m,,)' A" · A
- -
811"
Ii
(10.64)
•
= iJ" A' - iJ' A" must itself be modified. for
is excluded by local gauge invariance; as before. the gauge fields must be massless.
But this time the old association F'"
with this definition the gauge field Lagrangian (Equation 10.63) is not invariant
either (see Problem 10.12). Rather. we take'
F'"
"
A' _ o"A" - �(A" x A'l
E iJ
"
• This definition is not as arbitrary as it may
seem. The point s
i tlw with Ihru:-VKIor
fields there is .. second antisymmetric tensor
fonn available (A� x A'). and the coefficient.
-2qJ�. is chosen precisely IQ maL: it'll.
(10.65)
invariant. Notice that when the coupling
constant q goes to :terO we are left with the
free Dirac Lagrangian for each spinor field
and the free (massless) Prca Lagrangi.l.n for
each of the thrtt gauge fields.
10.4 ,(ang-Mills Thea",.
Under infinitesimal local gauge transformations (Equation 10.61),
FI<" -+ FI<" + �(). X FI<" )
'"
(10.66)
(Problem 10.13), and hence 2.... is invariant. (See Problem 10.14 for a proof that
the invariance extends tofinite gauge transformations.)
Conclusion: The complete Yang-Mills Lagrangian is
(10.67)
with FI<" defined by Equation 10.65; it is invariant under local SU(2) gauge
transformations (Equations 10.54 and 10.58), and describes two equal-mass Dirac
fields in interaction with three massless vector gauge fields. All this results
from insisting that the globa! SV(2) n
i variance of the original free Lagrangian
(Equation 10.46) shall hold Iocal!y. Borrowing the language ofelectrodynamics, we
say that the Dirac fields generate three curnnfs
(10.68)
which act as soun:es for the gauge fields; the Lagrangian for the gauge fields alone
2=
,
' F I<" . FI<> - -,I< . AI<
,
",
--
(10.69)
is reminiscent ofthe Maxwell Lagrangian (Equation 10.23), and gives rise to a rich
and interesting classica! field theory [7[ (see Problem 10.15).
Although Yang-Mills theory was inspired by the same idta as Weyl's (namely: a
global invariance should hold locally), the implementation was more subtle at two
points: (i) the local transformation rule for gauge fields, and (ii) the expression for
FI<" in terms of AI< . Both complications derive from the fact that the symmetry
group in question is non-Abelian (2 x 2 matrices do not commute, whereas 1 x
1 matrices - obviously - do). To emphasize the distinction, we refer to the weyl
case as an Abelian gauge theory and Yang-Mills as a non-Am!ian gauge theory.
In contemporary elementary particle physics, many symmetry groups have been
explored; we shall encounter a few in the remaining sections ofthis book. However,
the hard work is over: extending non-Abelian gauge theQry to higher symmetry
groups s
i a straightforward procedure, once the Yang-Mills model is on the table.
Curiously, though, Yang-Mills theQry in its original form turned out to be of
little use. After aU, it starts from the premise that there exist two elementary spin-�
particles ofequal mass, andas far as we know there are no suchpairs innature. Yang
and Mills themselves had the nucleon system (proton and neutron) in mind, and
thought of their model as a way of implementing Heisenberg's 5i0spin invariance
in the strong interactions. The small mass difference between proton and neutron,
1.29 MeV/,z, would be attributed to electromagnetic symmetry-breaking. For the
1 365
1661 10
GGuge Throriel
theory to succeed there had to exist a massless isotriplet ofvector (spin-I) particles.
The only candidates in sight are the p mesons; but they are hardly massless (Mp
= 770 MeV/(2), and this is not a minor discrepancy that can be plausibly blamed
on electromagnetic contamination. A number of attempts were made to doctor up
Yang-Mills theory to accommodate massive gauge bosons, but by the time they
finally bore fruit (through the Higgs mechanism) it was pretty clear that p, n, and p
are composite particles anyway. and that isospin is just one component ofa larger
flavor synunetry that is too drastically broken to play any fundamental role in the
strong interactions. When non-Abelian gauge theory finally came n
i to its own, it
was in the context of color SUP) symmetry in the strong interactions and (weak)
isospin-hypercharge SU(2)L ® Uti) symmetry in the weak interactions. Meanwhile,
for more than a decade after 1954 the Yang-Mills model languished - a lovelyidea
that nature had evidently chosen not to exploit.
10.S
Chromodynamics
According to the Standard Model, each flavor ofquark comes in three colors - red,
blue, and green. Althoughthe variousflavors carry different masses (fable 404), the
three colors of a given flavor are all supposed to weigh the same. Thus the free
liIgrangian for a particular flavor reads
(10.70)
As before, we can simplifythe notation by introducing
(10.71)
so that
(10.72)
This looks just like the original Dirac liIgrangian, only 1/1 now stands for a
three-component column vector (each element ofwhich is itselfa four-component
Dirac spinor). Just as the one-particle Dirac lagrangian (Equation 10.14) has (global)
Uti) phase invariance, and the (equal mass) two-particle Lagrangian (Equation
10041) admits U(2) invariance, so this (equal mass) three-particle Lagrangian ex·
hibits UP) symmetry. That is to say, it s
i invariant under transformations of the
form
(10.73)
where U is any unitary 3 x 3 matrix:
Ut U =: 1
(10.74)
But remember (Equation 10.50). any unitary matrix can be written as an
exponentiated Hermitian matrix:
with Ht =:= H
(10.7S)
Moreover. any 3 x 3 Hermitian matrix can be expressed in terms of nine real
numbers. �l. �2. . . . • a8. and (J (Problem 10.16):
H =:= (Jl + A · a
(10.76)
where 1 is the 3 x 3 unit matrix. Ai. A2 . . . .. A8 are the Gell·Mann matrices
(Equation 8.34). and the dot product now denotes a sum from 1 to 8:
(10.77)
Thus.
u d'.!' •
(10.78)
i the second
We have already explored phast transformations (eOl); what is /leW s
term. The matrix eil. · · has determinant 1 (see Problem 10.17); it belongs to the
group SU(3)! So what we are interested n
i
is the invariance of the lagrangian
(Equation 10.72) under SU(3) transformations. a global symmetry that we now
propose to make local.
That is. we modifY !Z in such a way as to render it invariant under local SU(3)
gauge transformations:
(10.79)
(again. I let ¢ !!! -(ticJq)a. with the coupling constant q playing a role analogous to
electric charge in QED). As always. the trick is to replace the ordinary derivative.
a" . by the 'covariant derivative' g,,,:
(10.80)
and assign to the gauge fields A" (there are eight ofthem notice) a transformation
rule such that
.
(10.81)
• III th� langu.ag� ofgroup theory. U{J)
.,
U(I) 0 SU(J).
368 1 10 w.ugt Thtorits
Again (see Equation 10.58), this entails
(10.82)
which, for the infinitesimal case, yields a formula identical to Equation 10.61:
(10.83)
However, this time the cross product notation is shorthand for
(B X C)i =
•
L f�It.BjCIt.
j.t_\
(10.84)
wherefijlt. are the structure constants of SU(3) (Equation 8.35), analogous to fiji<
for SU(2) (Problem 10.18).
The modified Lagrangian
Y = ihc"ifiyl'.:$10'" - mil"ifi'" = [ilic"ifiyl'iJl''" - mil"ifi"'l -
(q"ifiyI'A"') . AI'
(10.85)
is invariant under local S UP) gauge transformations (Equations 10.79 and 10.82),
but as usual the cost s
i the introduction of gauge fields AI' (eight of them, this
time). [n particle language, these correspond to the eight giuons, just as the U(l)
gauge field n
i Weyl's theory represents the photon: To finish the job, we must
adjoin the free gluon Lagrangian
I
Ygll>Onl = _ _
_ FI'" . FlO"
16rr
(10.86)
where, as in the Yang-Mills case
(10.87)
(with the SU(3) 'cross product' defined by Equation 10.84).
ColtClusion: The complete Lagrangian for chromodynamics is
(10.88)
Y is invariant under local SU(3) gauge transformations and describes three
equal-mass Dirac fields (the three colOrs ofa given quark flavor) in interaction with
eight massless vector fields (the gluons). It derives from the requirement that the
• Remembf,r th�t � 'ninth gluon·. coupling universcl1y to �ll quarks. is excluded by experiment
[see Problem 8.11).
10.6 F�y"mQ" Rules
global SU(3) symmetry of the original Lagrangian (Equation 10.70) should hold
locally. The Dirac fields constitute eight color currents
(lO.89)
which act as sources for the color fields (AI')' in the same way that the eketric current
acts as the source for eltctromagnttic fields. The theory described here is very close
in structure to that of Yang and Mills; in this case, however, we believe it to be
the correct description of a phenomenon realized in nature: the strong interaction.
(Of course, we need six replicas of 1/1, in Equation 10.88, each with the appropriate
mass, to handle the six quark flavors.)
10.6
Feynman Rules
Up to this point, the Lagrangians we have considered might just as well describe
classical fields as quantum ones; indeed, the Maxwell Lagrangian will be found in
any textbook on classical electrodynamics. The passage from a classical field theory
to the corresponding quantum field theory does not involve modification of the
Lagrangian or the field equations, but rather a reinterpretation of the field variables;
the fields are 'quantized: and partidts emerge as quanta of the associated fields.
Thus. the photon s
i the quantum of the electrodynamic field, AI'; leptons and
quarks are quanta ofDirac fields; gluons are quanta ofthe eight SU(3) gauge fields;
and W± and z!! are quanta of the corresponding Proca fields. The quantization
procedure itself is recondite, and this is not the place to go into it [8J; for our
purposes the essential point is that each Lagrangian prescribts a particular set of
Feynman rub. What we need, then, is a protocol for obtaining the Feynman rules
dictated by a given Lagrangian.
To begin with, notice that Sf consists of two kinds of terms: the free la·
grangian for each participating field, plus various intaaction terms (Zin,). The
former - Klein-Gordon, for spin 0; Dirac, for spin i; Proca, for spin 1; or
something more exotic, for a theory with higher spin - determines the propaga­
tor; the latter - obtained by invoking local gauge n
i variance, or by some other
means - determine the vertexfactors:
Free Lagrangian => propagator
Interaction terms => vertex factors
Let us consider the propagators first.
Application of the Euler-Lagrange equation to the free Lagrangian yields the
free field equations (Eqs. 10.13. 10.15, and 10.22):
[(11'(11' (';)2]
+
¢=0
(Klein-Gordon, for spin 0)
1 369
370
I JO
Gauge The'Hie,
[
( for spin D
[iyl' al' - (n;)] '" = 0
al'(al'A" - a'A") +
Dirac,
(n;f A'] = 0 (Proca for spin 1)
,
The corresponding 'momentum-space' equations are obtained by the standard
prescription PI' _ iliill':
IF - (me)lloP = 0
(spin 0)
(10.90)
11- (me)]", :: 0
sPin
(10.91)
(spin 1)
(10.92)
[(_p2 + (me)2}gJ.!' + PJ.!P.]A' = 0
( D
The propagator is simply (i limes) the invtm ofthe factor in square brackets:
Spin·O propagator:
Splll· l propagator
Spin.} propagat
or:
:;,,----;::(me}
::;;cl
(10.93)
_
_
_
'
= , ),[
1,, + me)
y
., _ me
...1.
y
i
(me)1
-
p2
(me)2
(10.94)
PlOP; 1
[&.. - (me)
(10.95)
Note that in the second case this factor is a 4 x 4 matrix and we want the matrix
inverse; in the third case the factor is a second·rank tensor (TJ.!') and we want
the tensor inverse (T-1)J.!" such that Tl'l(ll )�' :: 8; (Problem 10.19). These are
precisely the propagators we used n
i Chapters 6, 7, and 9: Since we obviously
cannot set m ...". 0 in the Proca propagator (Equation 10.95), we must go back to the
free field equation (Equation 10.22) to work out the photon propagator:
al'(a"A' - a"A") = O
(massless spin 1)
(10.96)
As I have remarked before, this equation does not uniquely determine AI'; if we
impose the Lorentz condition (Equation 7.82)
al'A" = 0
then (Equation 10.96) reduces to
alA" :: 0
• Actually. this proc�ur� only d�!�rmines the
propasalor up 10 a multiplicative constant.
since the field equalions can aIWlIYO t.. mul·
tipli� by such a factor. In the ·canonical·
fonn of these equations, the coefficient of
(10.97)
me (mc)l
or
is taken to be ± I, with !h� sisn
matchins that of the
in Z. Other
conventions lead to a slilJb!1y diffe",nt set of
Feynman rule-<, but do no!, of course. cban�
the C1lcub!� reaction amplitudes.
mus lerm
10.6 Feynmon Rules
which, in momentum space, can be written as
(10.98)
So the photon propagator is
Massless spin-l propagator.
(10.99)
To get the vertex factors, first write down i Yi"' in momentum space (iM", -+ p",)
and examine the .fields involved; these determine the qualitative structure
of the interaction. For example, in the case of the QED Lagrangian (Equation
10.35)
(10.100)
there are thru fields involved (�.1Jr, and A",) and this defines a vertex in which
three lines are joined - an incoming fermion, an outgoing fermion, and a photon.
To obtain the vertex factor itself, simply rub out the.field variables:
(QED vertex factor)
(10.101)
(In the case of the photon, what we actually rub out is ./7fC]4ifA",; the extra factor
is due to our use of cgs units which are, for this purpose, a little cumbersome.) lbe
same goes for chromodynamics (Equation 10.88): the quark-gluon coupling
(10.102)
yields a vertex ofthe form
9
q
with the vertex factor
(10.103)
(Ibe strong coupling constant is traditionally defined with a factor of 2: g, ;;;;
2..j4ii"J1fiq, where q is the 'strong charge· appearing in the Lagrangian). However,
there are also direct gluon-gluon couplings, coming from the F"'V . F,," term in y,
1 311
372 1
10 Clluge Tmories
since P" contains not only the 'free' part, 01'
0" but also an interaction
term, - 2qjlU:(A'"
V) (Equation 10.87). Squaring it out, we find
A' - AI' ,
)( A
-) ((aI'A" - ,VAl') (A", )( Av) + (AI' )( A") . (a"A, - a.A,,)]
= (8,,,,
(10.104)
- 4 ) (N' AV) . (A", )( A,)
'
i
x
Jr lU: 1
AI',
The first term carries three factors of
and leads to the three·gluon vertex
the second term carriesfour factors of
and gives the four-gluon
(Equation
vertex (Equation 8.43). (For practice in extracting Feynman rules from Lagrangians,
see Problems 10.20 and 10.21.)
8.42);
A"',
10.7
The Mass Term
The principle of local gauge invariance works beautifully for the strong and
electromagnetic interactions. In the first place, it gives us a machint for determining
the couplings (in the 'old days' the construction of2'inl was a purely ad hoc guess).
i the early 1970s, [91 gauge theories are
Moreover, as 't HooR and others proved n
renormalizable. But the application to weak interactions was stymied by the fact
that gauge fields have to be massless. Remember, the mass tenn in the Proca
Lagrangian is not locally gauge invariant, and while the photon and the gluons
ZO certainly are not. So the question arises: can we
doctor up gauge theory in such a way as to accommodate massive gauge fields?
The answer is yes, but the procedure - exploiting spontaneous symmetry-breaking
and the Higgs mechanism - is diabolically subtle, and it pays to begin by thinking
very carefully about how one identifies the mass term in a Lagrangian.
Suppose, for instance, you were given the following Lagrangian for a scalar
field
arc massless, the W's and the
¢:
:t' ""
!(a",¢)«l"ofJ
) + C-(o-.;)l
2
(10.105)
where (f is some (real) constant. Where is the mass term here? At first giance
there's no sign of one, and you might conclude that this is a massless field. But that
is incorrect, for if you expand the exponential, !i' takes the form
1
1 6 6
!i' "" z(a",
¢)«l"4» + l - ct2¢2 + "21 (f44>. - 6()'
¢ + ...
(10.106)
The 1 is irrelevant (a constant term in !i' has no effect on the field equations), but
the second term looks just like the mass term in the Klein-Gordon Lagrangian
(Equation 10.11), with
= !(mc/Ii)2. EVidently. this Lagrangian describes a
particle of mass
ct1
m=J
2aIi/c
(10.107)
10.7 Th� Moss Term
The higher-order terms represent couplings, of the form
,
,
,
/
/
/
�
"
,
/
,
/
and so on, This s
i not supposed to be a realistic theory, ofcourse - I offer it only as
an example ofhow the mass term in a Lagrangian may be 'disguised'. To expose it,
we expand Z in powers of
tP and pick out the term proportional to </J2 (in general,
it's the term of second order in the fields - </J, 1/1 , N', or whatever),
But there is a deeper subtlety lurking here, which [ illustrate with the following
Lagrangian:
(10.108)
Here J.! and ). are (real) constants. The second term looks like a mass (and the
third like an n
i teraction). But wait! The sign is wrong (compare Equation 10.11) - if
that's a mass term, then m s
i imaginary, which is nonsense. How, then, should
we interpret this Lagrangian?' To answer this question, we must understand that
the Feynman calculus is really a perturbation procedure, in which we start from
the ground state (the 'vacuum') and treat the fields as fluctuations about that
state. For the Lagrangians we have considered so far, the ground state - the field
configuration of minimum energy - has always been the trivial one: </J "" O. But
for the Lagrangian in Equation 10.108, </J
0s
i nol the ground state. To detennine
the true ground slate, we write .!l' as a 'kinetic' term
minus a 'potential'
=
(!a,..tPi"'</J)
term (inspired by the classical Lagrangian, Equation 10.4):
(10.109)
and look for the minimum of%' . In the present case,
(10.110)
, [ lik� to im.ogi"" that God has a giant
comput�r-controUed r..ctory. which takes
lagrangian.. as input and ddi�rs tht uni·
�rses they r�present as output. Usually God's
computtr has no difficulty - whtn you fttd
in the Maxw�ll lagr..ngian. Equation 10,35.
for aample, it irrunediatdy creates an dt<:­
trom;ogn�tk uni�rse or interacting electrons.
positrons, and photons. Sometimts it tak�s
a lillIe longer _ th� lagrangian in Equation
10.1OS. for instance. confuses it at first. unti
l
it deciphfis the 'hidden' mass term. And
occasionally it returns an errOr messa�;
'this lagrangian does not dtscrib.. a possible
universe: please check for synta. errOrs Or
incorrect signs'. ThaI's what it would do,
for example, ifyou fed h th� lagrangian in
Equation 10.108 without the J.. term.
1 373
374
J 10 Gouge Th,ories
and the minimum occurs at
� = ±/J-/>"
(10.111)
(see Figure 10.1). The Feynman calculus must be formulated in terms ofdeviations
from one or the other of these ground states. This suggests that we introduce a new
field variable, 1/, defined by
(10.112)
In terms ofI), the Lagrangian reads
(10.113)
The second quantity is now a mass term with the comet sign, and we discover
(comparing Equation 10.11) that the mass of the particle is
(10.114)
Meanwhile, the third and fourth tenns represent couplings of the form
,
,
,
/
/
/
�
"
,
/
,
/
(the last tenn is a constant, signifying nothing).
[ emphasize that these Lagrangians (Equations 10.108 and 10.113) represent
exactly tlu same physical syskrn; all we have done is to change the notation
(technically, a perturbation series in � would not converge, because it is an expan­
sion about an unstable point); only in the second formulation can we read offthe
mass and the vertex factors.
(Equation 10.112). But the first version is not suited to the Feynman calculus
•
Fig. 10.1 Graph of'!t(4') (Equal;on 10.110).
10.8 Spanlonro�, Symmerty-breoking
state (the field configuration for which %'(4)) is a minimum) and re.express '£ as
ColtClusion: To identify the mass term in a lagrangian, we first locate the ground
2
obtain the mass from the coefficient ofthe '1 term.
a function of the deviation, I), from this minimum. Expanding in powers of I), we
10.8
Spontaneous Symmetry-breaking
The example we have just considered illustrates another phenomenon of impor'
tance: sponlanrous symmttry.brtaking. The original lagrangian (Equation 10.108)
is tIItn in r/!: it is invariant as r/! -.. -r/!. But the reformulated lagrangian
(Equation 10.113) is not even in '1: the symmetry has been 'broken'. How did
this happen? It happened because the 'vacuum' (whichever of the two ground
states we care to work with) does not share the symmetry of the lagrangian. (The
collection of aU ground states, of course, does, but to set up the Feynman formalism
we are obliged to work with one or the other of them, and that spoils the sym·
metry.) We call this 'spontaneous' symmetry-breaking because no txternal agency
is responsible (as occurs, for example, when gravity breaks the three-dimensional
symmetry in this room, making 'up' and 'down' quite different from 'left' and
'right'). To put it the other way around, the true symmetry of the system is 'con·
cealed' by the arbitrary selection ofa particular (asymmetrical) ground state. There
are examples of spontaneous symmetry-breaking in many branches of physics.
Take, for instance, a thin plastic strip (say, a short ruler): ifyou squeeze the ends
together, it will snap into a curved configuration, but it can just as well buckle to the
left as to the right - both are ground states for the system, and either one breaks
the left-right symmetry (Figure to.2).
But the spontaneously broken symmetry we have just considered was a discrete
symmetry, with just two ground states. More interesting things happen when
we consider continuous symmetries. (Replace the plastic strip in Figure 10.2 with a
,
,
,
,
,
,
fig. 10.2 Sponl�n�ous symmdry·b,e�king in a pl�slic strip.
1 315
3761 10 COllIe Theories
plastic rod - say, a knitting needle. lben it can buckle in �ny direction. not just
left or right.') It is easy to construct a Lagrangian with spontaneously broken
continuous symmetry. For example,
.!L' = t(il,,4>d(il"4>d + !(il,,4>21W4>2) + t�2 (4)f + ¢il - !>.. (4)f + 4>i)
l
2
(10.lIS)
lbis is identical to Equation
except that now there are two fields, 4>1 and ¢2,
and because :t' involves only the sum ofthe squares, it is invariant under rotations
in 4>1, 4>2 space.t
This time the 'potential energy' function is
10.1OS,
(10.116)
and the minima lie on a circle of radius !1-/>":
(lO.tt7)
10.3).
(Figure
To apply the Feynman calculus, we have to expand about a particular
ground state ('the vacuwn') - we may as well pick
(to.118)
Fig. 10.3
The potential function (Equation
10.116).
• A more sophistiQled example is the ferromasnet in the ground s�te �Il the electron spins are
alisned. but the din,l;'m of a1ignmem is an occident of history. The Ihtory s
i symmetriQI. but
a given piece of iron has to pick a pa"ku�r orien�tiQn. and that fspontane<lusly'l breaks the
symmetry.
t Group theoretiQlly. it is n
i variam under SO(2):�, ..... ¢" CO$ (J + �, sin (J; ¢'I
-¢" sin (J +
¢'l cos (J, for any 'rotation ansle' (J (see Problem 4.6).
-->
IUS Sponhmeous Symm�try-b(eokj"g
As before, we introduce new fields, '1 and �, which are the fluctuations about this
vacuum state:
1) 5! <Pl - J.l./)... ;
� 5! <f!2
(10.119)
Rewriting the Lagrangian in terms of these new field variables, we find (Prob.
lem 10.22):
Ie =
[�(a"I))WI))
] U(a,,�)(a"n]
- J.l.21)2 +
(10.120)
The first term is a free Klein-Gordon Lagrangian (Equation 10.11) for the field I),
which evidently carries a mass
(10.l21)
(same as before, Equation 10,114); the se.::ond term is a free Lagrangian for the
field �, which is evidently mass1=:
(10,122)
and the third term defines five couplings:
(the final constant, of course, is irrelevant). In this fonn the Lagrangian doesn't
look symmetrical at all; the symmetry of Equation 10.115 has been broken (or
rather, 'hidden') by the selection of a particular vacuum state.
The important thing to notice here is that one of the fields (�) is automaticaUy
masslt:SS. This is no accident. It can be shown (Goldstone's theorem [10]) that
spontaneous breaking of a continuous global symmetry s
i always accompanied
by the appearance of one or more massless scalar (spin·O) particles (we call
them 'Goldstone hosans').· Well, this is a disaster; we were hoping to use the
me<hanism of spontaneous symmetry.breaking to account for the mass of the
weak interaction gauge fields, but now we find that this n
i troduces a massless
i no such thing on the roster of known elementary
scalar boson, and there s
particles.t But hold on, for there is one final incredible twist in the story. It comes
• Intuitively, this is Te�ted to the f�ct tim !here is no rWSI�nce (0 e�dmions in the � direction.
Flick th� bent huning needle and il will spin fr�ly about th� axis, wh�r�as r<>ditJt excitations
encounter a restoring for«, and the system osciU�tes.
t 11 is hard to imagine th�t such a �rtide could h�ve escaped detection, With k",,"Y �Ttides. this
is �Iway$ a ponibility _ maybe you just didn't have �nough energy 10 produu il - but a I"I"I<W­
!ess particle would surely have sllown up sotmWhere, if only in the form of 'mining' energy �nd
mo=nlum.
I
J71
Hal 10
Gouge Thwries
when we apply the idea of spontaneous symmetry.breaking to the case of local
gauge invariance.
10.9
The Higgs Mechanism
The Lagrangian we studied in Se<tion 10.8 can be written more neatly if we
combine the two real fields, 11 and 12, into a single complex field:
(10.123)
so that
(10.124)
In this notation (and it is nothing but notation), the Lagrangian (Equation 10.115)
reads
(10.125)
and the rotational S0{2) symmetry that was spontaneously broken becomes invari·
ance under U(l) phase transformations:
(10.126)
This is precisely the kind of symmetry we considered back in Section 10.3, except
that now we are working with scalar fields instead of with spinors. We can make
the system invariant under local gauge transformations
(10.127)
by the usual device of introducing a massless gauge field A", and replacing the
derivatives in Equation 10.125 with covc.zriant derivatives (Equation 10.38):
(10.128)
(10.129)
10.9 The Higgs Mech(i/1ism
Now we simply retrace our steps in Section
invariant Lagrangian (Equation
Defining the w fie d
10.8. applying them to
ne
l s
10.129).
the locally
(10.130)
Lagrangian
z
U(il"'I)(il"'I) �2tl] [�(a,,�)(il"�)]
+[ �
� (t ir A"A"]
{!(I) (il,,�) -�(il"I)))A" + i (tf I)(A"A")
+ i ( f (�l + 1)1) (A A") -.l.;.t(I)} + 1)�2)_ �.l.2(1)4
1�2 + �4)}
t
"
+ (�.i) I '",)A' ( )
.I. fie
2.1.
fir line is
same as
In The second line describes
(I)). of ..fi�ti./c. nd
the free
it has acquired a s
A". but
(compare Equation 10.119). the
becomes (Problem 10.2S):
+
-
=
-
F F
1 1T "" ,," +
+
+
The
st
the
mass
a
gauge field
+ 21)
' '
"
(10.131)
before (Equation 10.120): it represents a scalar particle
a massless Goldstone boson
mas :
- mirabik dictu! (10.132)
the
Equation
term in curly brackets
various couplings of" �. 'I. and A"
is i eres i g
see where the mass of A came from:
(Equation
contains
of the form A"A". which
r pr s n a
(compare
10.121). The
Proca Lagrangian.
specifies
(Problem 10.26). It
nt
tn
to
the original Lagrangian
10.129)
a term
</1*</1
- absent spontaneous
symmetry-breaking - would e e e t coupling:
V
/
/
/
/ ¢
/
,
,
,
¢ "'
,
But when
'off and
picks up a constant
(Equation
this of Lagrangian emerges as a Proca
However. we still have unwanted
boson (n. Moreover. there a
suspicious-looking quantity in Z:
the ground state moves
10.130).
piece the
that
center'.
the field </11
mass term.
Goldstone
is
(10.133)
1 )19
380
I 10 Gauge Theories
What are we to make of fhis? Ifwe read it as an intuaction, it leads to a vertex of
the form
,
- - - -
A
�
in which the � turns into an A. Any such term, bilinear in two different fields,
n
i dicates that we have incorre.::tly identified the fundamental particles in the
theory (see Problem 10.23). Both difficulties involve the field �
4J2 , and both
can be resolved exploiting the local gauge invariance of.2' (in the original form,
Equation 10.129) to transjonn this fidd away entirely! Writing Equation 10.126 in
terms ofits real and imaginary parts,
""
4J -+ 4J' = (cos O + isinO)(4JI + i4J2)
= (4Jl cos (J
we see that picking
-
<h. sin (J) + i(¢>l sin(J + ¢>2 cos tI)
(10.134)
(10.135)
will render ¢>' real, which s
i to say that ¢>'2 O. The gauge field A" will transform
accordingly (Equation 10.34), but the lagrangian will take the same form in terms
of the new field variables as it did n
i terms of the old ones (that's what it /mans to
say that !£ is invariant). The only difference is that � is now zero. In this particular
gauge, then, the Lagrangian (Equation 10.131) reduces to
=
:t'
:: [�(a"I))(a"I)) - #21)2] + [- l�l/"·Fl'. � (-� tfY A"AI']
{i (�)2 I){A"A") + � (-�f I)2(A"AI') - 1-#1)3 - �I-ll).}
+
+
(10.136)
By an astute choice of gauge, we have eliminated the Goldstone boson and the
offending term in :t'; we are left with a single massive scalar (the 'Higgs' particle)
and a massive gauge field A".
Please understand: The Lagrangians in Equations 10.129 and 10.136 describe
exactly the same physical system; all we have done s
i to sele.::t a convenient gauge
(Equation 10.135) and rewrite the fields in terms of fluctuations about a particular
ground state (Equation 10.130). We have sacrificed manifest symmetry in favor
of notation that makes the physical content more transparent, and allows us to
extract the Feynman rules more directly. But it's still the same Lagrangian. There
is an illuminating way to think about this: a masslt:SS vector field carries two degrees
of freedom (transverse polarizations): when AI' acquires mass. it picks up a third
(longitudinal polarization). Where did this extra degree of freedom come from?
Answer: It came from the Goldstone boson, which meanwhile disappeared from
I)
10.9 The Higgs Mechanism
the theory. lbe gauge field 'ate' the Goldstone boson, thereby acquiring both a
mass and a third polarization state.* lbis is the famous Higgs mechanism, the
remarkable offspring of the marriage of local gauge invariance and spontaneous
symmetry-breaking [11].
masses of the weak interaction gauge bosons (W:!: and ZO). The details are still
According to the Standard Model, the Higgs me<:hanism is responsible for the
matters of speculation - the Higgs particle has never been seen in the labora·
tory (presumably it is just too heavy to make with any existing accelerator), and
��2(4)'4>) + �).2(4)'4>)2
the Higgs 'potential', 0/.1(4)), is unknown (I used %' = _
just for the sake of argument).t There may in fact be many Higgs particles,
or it may be a composite struchlre, but never mind: the m
i portant thing is
that we have found a way in principle of imparting mass to the gauge fields,t
and that is our license to believe that all the fundamental interactions - weak
as well as strong and electromagnetic - can be described by local gauge theo·
ries [12J.
References
1 For �n introduction to 1M Lt·
gr�ngi.m formul�tion of clasi·
011 mechanics, see Taylor, ). R.
(2005) Classical Mechanic., Uni·
versity Science Books, Sausalito, C.A. or; (a) Marion. ,. B. and
Thornton. S. T. (1995) Classical
Dynamic.< afParlick, "nd SY'Ums,
�th edn, Harcourt, Orlando, F.L
2 The Euler-lagrange equ;otion
can � obbined, in turn, from
the Prn
i ciple of Least Action.
See Poole, C. 1'., Sapko, ,. L. and
Goldstein. H. (2002) Classical Mt·
chanies. 3rd edn, Addison·Wesley,
Reading. MA or; (a) lanczos. C.
(1986) 1M Variational Principln
,:,fMu;hanics, Dover, New York.
• We don't have to adopt any particu�r gauge.
H�..", f
i we de not, the t� will contain
a nonphysiCoil 'ghost' particle, and it is sim·
plest to eliminate it elIplicitly from the start.
t Acmally. for the theory to be renormalizable
the potenti;ol has to � quartic in the fields.
t In the Stancbrd Model, the Higgs parti.
cle is also responsible for the masses of
q�rks and leptons; th..y are initi;olly taken
to be massless, but are assumed to ha.�
Yub..... couplings (""" Problem 10.21)
ro the Higgs particle. When the �tI.." is
'shifted' by spontaneous symmetry·breaking
3 Weyl. H. (1919) Annals af PhY5ic.<,
59, 101. See also; (a) A1·Kuwari,
H. and Taha, M. O. (1991) Amtr·
ican Journal ofPhysic., 59, �. or;
(b) Moriyasu, K. (1981)
l·
ementary Primerfor Gauge 1M.
aI}', World Scientific, Singapore.
An E
• Yang. C. N. and Mills, R. L. (1954)
Phy.it;, Revit;..., %, 191. For histor·
kal commenbry see; (a) Mills, R.
(1989) Amtrican Jo..rnal of Physic.<.
51, �93. or; (b) 't Hoofi, G. (2005)
Fifty Yoar:! of Yang-Mias 1Mary.
Singapore: World Scientific.
S See. for example, Chevalley. C.
(19.f6) TheaI}' afLie Gro"p5. Prince­
ton University Press, Princeton. N.).
(Equation 1O.130J, the Yub..... coupling
splits into two parts. one ofwhich is a true
intenction and the other a "'= term for
the field .;. This is a nice idta, but it does
00( help us to � the fermion ma�s.
since the Yub..... coupling constants them·
selves are unknown. Dilly when (and if)
the Higgs particle is actuallyfou"d win it be
possible to confirm all this empirically. Scr
Chapter 12.
1381
3321 10 Gouge
Theories
IlIttractiollS, World Scientific,
6 Local gauge th(or;(s are treated
with ex\r.iIordinuy clarity and �iUty
in the unpublished research notes
Wheeler, N. A. (1981) 'Oomiclll
Sio�pore,
10 Goldstone , I . (1961) Nllovo G·
""'lila, 19, 1504. Goldstone, I.,
Salam, A. and Weinberg, S.
Chromodynamic,' �nd '8�rt Bones
oflhe Classic.aJ Theory of Gauge
(1962) Phy,lics Review, 127, %5.
11 Higgs, P. W. (19(4) Phy,lics u:tltt'3,
7 A<:tor, A. (1979) Revirwo ofModern
12, 132; Englert, F. and Brout, R.
(1964) Ph)'Sicai Review Wltn, 13,
321.
Fidds', Reed College, Portl,md, O.R.
Phy,lic" 51, 46\.
5 The details will � found n
i any trea·
tise on quantum field theory, such
as those cited in the Introduction.
9 'I Hooft, G. (1971) NuclI:ar Phy,lics,
B33, 173; (1971) B3S, 167. Reprinted
in (a) Lai, C. H. led) (1981)Gauge
Theory of Wtak and Eltaromagnelic
12 For detailed applkation of the Higgs
mechanism in the electroweak the­
ory of Glashow, Weinberg, and
Salam, see, for instance, Chapter
IS of Halun, F. and Martin, A. D.
1984) Quarks and Ltpu>lIs John
Wiley & Sons, Ltd, New York.
Problems
10.1 One advaOlage of the l.:Igrangiao formulation is that it does not commit us to any
particular coordinate system - the q's in Equation 10.6 could be Cartesian coordinates,
Or polar coordinates, or any other variables w<: might use to designale the particle s
position. Suppose, for example, we want to analyze the motion of a particle that slides
frictionlessly on the inside surface ofa cone mounted with its axis pointing upward, as
shown,
'
,
y
x
(a) Express T and U in terms ofthe variables .z and � and the consunts a (the opening
angle of the cone), m {the mass of the particle}, and g (the acceleration ofgravity).
(bt Construct the l.:Igrangian, and apply the Euler-Lagrange equations to obtain differ·
ential equations for z(l) and � (I).
(e) Show that L = {m tan�a).zl¢ is a constant of the motion. What is this quantity,
physicaJly�
10.9 Th� HiW Med!(mi!m
(d)Use the result in (e) to eliminate 4> from the z �u;ation. (You are left with a
=nd-order differential equation for z(l): if you want to pursue the problem further,
it is easiest to invoke conselVation of tlttlg}'. which yields afirn-order equation for :.)
10.2 Derive Equation 10.17
10.3 St4lrting with Equation 10.19, show that 3"A"
0, and hence that each component of
A" satisfies the Klein-Gordon equation: DA" + (me/h)lA" _ O.
10.4 As it st4lnds, the Din.c Lagrangian (Equation 10.1-4) treats '" and f asymmetrically. Some
ptople prefer to deal with them on an equal footing. using the modified Lagrangian
=
this 2. and show thatyou get the Din.c�uation
(Equation 10.15) and its adjoint.
10.5 The Klein-Gordon Lagrangian for a compkx field would �
Apply the Euler-Lagrange equations to
Treating 4> and </>* as independent field variables, deduce the field equations for each.
and show that thf,se field �uations are consistent (i.e. one is the complex conjugate of
the other).
10.6 Apply the Euler-Lagn.nge equations to Equation 10.33 to obtain the Dine equation
with electromagnetic coupling.
10.7 Show that the Dirac rurrent (Equation 10.36) satisfies the continuily equation
(Equation 10.25).
10.& The complex Klein-Gordon Lagrangian (Problem 10.5) is invariant under the global
gauge transformation 4> ..... .;fI4>. Impose loc;ol gauge invariance to construct the
complete gauge·invariant Lagrangian. and determine the current density r. Using the
Euler·Lagn.nge �uation for 4>, show that this current obeys the continuity equation
(Equation 10.2S). (Warnil1g: The cwrent is defined by Equation 10.H. ItO! by Equation
10.23. It is true that the former follows (ordinarily) from the latter, but not when 1"
n
i 2 that is proportional to A": rather, you must use the Euler-Lagn.nge equations to
depends explicitly on A". In this (rare) cireumst4lnce you cannot just pick offthe term
de�rmine 3" P", and get the current from that.)
10.� (_I Suppose the field variables (4)i) are subjected to an infinitesimal global transformation
8</>;. Show that the Lagrangian .ft'(4)i. 3,,4>;) changes by an amount
In particular, if the Lagrangian is iltvarialtl under the transformation in question,
then !2 = O. and the term in curly brackets constitutes a conserved rurrent (that is, it
obeys the continuity equation). More precisely, if the transfomution �i is specified
by a pan.meter !8. the Noetherian current is
(up to an overall constant, chosen for convenience in the particular context). This
is the essence of Noether's theorem (3), relating symmetr�s of the Lagrangian to
conservation laws.
1 3&3
33<41 10
GOIIBe Theories
(1)1 Apply Noether's theorem to � Oir.lc LaBr.In�n (Equation 10-14), to construct the
conserved current associated with global pluseinvariance (Equation 10.26). Comp,aff
the electric current (EqUOItion 10.36).
Ie) Do the same for the complex Klein-Gordon Lagr.lngian in Problem 10.8
10.10 Derive EqUOItion 10.51
10.11 !}duce Equation 10.58 from Equation 10.57, using Equations 10.�-10.56.
10.12 Suppose we were to define
in Yang-Mills theory.
(>] Find the tr.lnsformation rule for this P". under infinitesimal �uge transformations
(Equation 10.61).
(tt) Determine the infinitesimal tnnsformation rule for .:t'll (Equation 10.63), in this
case. Is the Lagrangian invari
ant)
10.11 Derive EqUOItion 10.66, sUlrting with Equations 10.61 and 10.65.
10.14 Prove that gauge field Lagr.lng�n (Equation 10.63) is invariant under finite local gauge
transformations. as follows:
(a) Using Equations 10.58 and 10.65, show that
[Note that a�(S-LS) == O ;j. (a� S-I)S = -S-I{a�S).)
(hi Show. therefore, that
Tr[(y . F�")(Y . F".)]
is invariant.
(e) Using Problem 4.20(c). show that the trace n
i (b) is �ual to 2F'" . F�•.
10.15 Apply � Euler-Lagr.lnge equations to the Lagrangian in Equation 10.69. Using the
standard associations (Equations 7.71, 7.72. and 7.79), obtain 'Maxwell's eqUOltions' for
classical Yang-Mills theory. [Note that there are ,II"", charge densities, three current
densities, three scalar potentials. three vector potentials, three 'electric' fields. and three
'magnetic' fields. in this theory.) (Unlike electrodynamio, your expressions for the
divergence and curl of the E's and B's will inevitably nvolve
i
the pot�n'iaIs.)
10.16 Show !lut any Hermitian 3 x 3 matrix can be written as a linear combination of �
unit matrix and the eight Gell·Mann matrices (EqUOltion 10.76)
10.11 (a) Show that det(�) == ,TT{I\). for any matrix A. [Hint: Check it first for a diagonal
matrix. Then extend the proof to any diagonaUzobk matrill (S-I AS == D. where D
is diagonal, for some matrix S) - show !lut Tr(A) == Tr{O) and S-I .... S = �D. so
that det{....) = det{�� . Of course, not all matrices an diagonalizable; however, every
matrix can be brought n
i to Jordan canonical form {S-'AS ==], where ] is diagonll.!
except for some l's immediately below the main diagonal). nke it from there.j
(b) Show that tfl·, (in Equation 10.78) has determinant 1.
10.18 Starting with Equation 10.81, derive Equations 10.82 and 10.83.
10.9 Th� Higss Mu/wnism
10.19 Confirm th;ot the PrOQl prop;lg�tor (Equ�tion 10.95) is the n
i verse of the tensor in
Equation 10.92, in the sen� explained in the text.
10.20 Construct the ugrangi�n for ABC theory (Ch;opter 6).
10.21 Give � physical int�rpreUlion ofthe Yuhwa ugnngian:
:t'
=
lilictyl' 0l'tP - mlc2V/1/r1
+
[�(Ol'oiI)(i:II'oiI) � (m;')l oiI2] ayl1/roil
-
-
(10.137)
What are the spins �nd mas�s of the p;lrticles? Wlut ue their prOp;lg<ltors? Dr�w the
i
Feynnun diagnm for their nteraction
and detennine the vertex f;wor.
10.22 Derive Equation 10.120
10,23 Suppose we took
�s
the fundamental fields. instead of Equation 10.119 Express the ugrangian
(Equation 10,120) in tenns of >{I, and >{II.
(Commrnl: Offh�nd, it looks n though we have two mimi"" fields here, �nd thus escape
Goldstone's theorem. Unfortunately. there is also a term ofthe form _".1>{I1'/12. [fyou
interpret this �s �n inkradion. it converts '/I I into "'I. �nd vice vers�, but th�t means
neither one exists as an independent free p;lrticle. Rather. such an expression should be
interpreted as an off·(liagonai term in the mass matrix (Equation 10.45). indicating th;ot
we have incorrectly identified th� fundamental fields in th� theory. The physical fields
are those for which M is diagonal and for which no direct transitions from one to the
other can occur. We have encountered this situation once before, in Section 4.4,3: we
found th;ot.l«l .... K'. and h�nce that these are not the physical p;lrtide states; inste�d,
the linear combin�tions K, and Kj, in terms ofwhich the mass matrix is di<lgon�l, ue
the 'true' p;lrticles.]
10.24 Generaliu th� ugument following Equation 10.115 to thru fields (4'" 4'2, 4'1). What ue
the mas�s of the three p;lrticles? How many Goldstone basOIIS are there in this ca�?
10.lS Starting from Equations 10.129 and 10.130. d�rive Equation 10.131
10.26 Oraw the primitive vertices for ;ill the inler�ctions in curly brackets in Equation 10.131
Cirde the ones that survive in Equation 10.136
I38S
11
Neutrino Oscillations
Rea"t txptrimtnts have shown that �utrin05 can convertfrom onejlullOrto ,,"otner (for
instanct, v• ..... vJ.I)' This means that neutrinos havt: nonuro mass, and that the lepton
numbers (ekelron, muon, and tau) an: not separately conserved. Neutrino oscillations
resolve Jm sclar neutrino problem, lind suggest mcdtst changes in the Standard Model.
The treatmmt hae is /argdy self-wntained. and could evt:n be rcad immediately after
Chapter 2.
11.1
The Solar Neutrino Problem
The siory begins [1) in the middle of the nineteenth century. when Lord Rayleigh
undertook to calculate the age of the sun. He assumed (as everyone did, at the
time) that the source of the sun's energy was gravity - the energy accumulated
when all that matter 'fell down' from infinity is liberated over time in the form
of radiation. On the basis of the known rate of solar radiation (which he took to
be constant), Rayleigh showed that the maximum possible age of the sun was
substantially shorter than the age ofthe earth as estimated by geologists, and, more
to the point, shorter than Dazwin's theory of evolution required. This pleased Lord
Rayleigh, who was opposed to evolution on quaint religiOUS grounds, but it worried
Darwin, who removed his own estimate from subsequent editions ofhis book.
In 1896, Becquerel discovered radioactivity. In subsequent studies he and the
Curies noted that radioactive substances such as radium give off prodigious
amounts of heat. This suggested that nuclear fission, not gravity, might be the
source of the sun's energy, and this would allow for a much longer lifetime. The
only trouble was that there didn't appear to bt any radioactive stuff n
i the sun,
which is made almost entirely ofhydrogen (plus a small amount oflight elements,
but certainly not uranium or radium).
By 1920, Aston completed a series of meticulous measurements of atomic
weights, and Eddington noticed that four hydrogen atoms weigh slightly more than
one atom ofhelium-4. This implied (in view of Einstein's E _ mel) that the fusion
offour hydrogens would be energetically favorable, and would release a substantial
J,Uroo:/UC!;OIlIO EltIlltIl!�'Y PGrlicW. Stcolld Ed;!;",,- D.vid Griffiths
Copyright Cl 2008 WILEY·VCH Verlag GmbH & Co. KG.". Weinheim
ISBN, 978-3·527·40601·2
388 1
"
N�lJtriflo (N;;!lot;o",
amount of energy. Eddington suggeste<! that this process (nuclear fusion) powers
the sun, and in essence he was right. Of course, Eddington didn't know what the
memanism for binding the hydrogens together might be; this had to await the
development of nuclear physics in the 1930s - in particular, Chadwick's discovery
ofthe neutron and Pauli's invention of the neutrino.
In 1938, Hans Bethe worked out the details, which tum out to be quite com·
plicated. In heavy stars the dominant mechanism is the CNO (Carbon-Nitrogen
-Oxygen) cycle, in which the fusion process is 'catalyze<!' by small amounts ofthose
three elements. But in the sun (and other relatively light stars) the dominant route
is the so-called pp chain (Figure 11.1). To begin with, a pair of protons (hydrogen
nuclei) combine to make a deuteron, a positron, and a neutrino. (The deuteron is a
proton and a neutron, so what really happene<! here is that a proton converted into
a neutron, a positron, and a neutrino - the reverse ofneutron decay.) Alternatively,
the outgoing positron could be replaced by an incoming electron. Either way, we
have produce<! deuterons (along with some neutrinos) from protons. The deuteron
soon picks up another proton to make a helium·3 nucleus (two protons and a
neutron), releasing energy in the form of a photon. Helium·3 has three options:
it can join with another loose proton to make an alpha particle - the nucleus of
helium-4 (two protons and two neutrons). Once again. a proton has been converted
into a neutron (with emission of a positron and a neutrino). Or two helium·3s can
get together to make an alpha particle and two leftover protons. Or the helium-3
can combine with an alpha particle (produced in one of the previous reactions) to
make beryllium.7, with the emission of a photon. Finally, the beryllium can either
The pp Chain
Step 1: Two protOIiS make a deuteron
Step 2: Deuteroll plus proton makes lHe.
d + p .... lHe+1"
Step 3: Hc!ium·3 make.. alpha partidc or 7[k
JHc+ p
3He+ 3He
lHe + a
o + e'" + v_
a +p+p
lB e + "I
Step 4: BcriUiUl" lHakes alpha par�icles.
7[3c + e�
lLi + v.
7Li + P
a+.
'[3c + P
88 + "1
"Be-
0+0
"B
aBe' + e� + v<
Fig. 11,1 The pp chain: how protons make alpha particles in the sun.
" _ , The SoIQr N�lJlrino Problem
absorb an electron, making lithium, which picks up a proton, yielding two alpha
particles, or else it absorbs a proton, making boron, which goes to an excited state
ofberyllium-S, and from there to two alpha particles.
The details are not so important; the point is that it all starts out as hydrogen
(protons), and it all ends up as a particles (helium-4 nuclei) - precisely Eddington's
reaction - plus some electrons, positrons, photons . and neutrinos. But is this
complicated story really Ime? How can we tell what is going on inside the sun?
Photons take athousand years to work their way out from the center to the surface,
and what we see from earth doesn't tell us much about the interior. But neutrinos because they interact so weakly, emerge virtually unscathed by passage through the
sun. Neutrinos, therefore, are the perfect probes for studying the interior ofthe sun.
In the pp chain there are five reactions that yield neutrinos, and for each one the
neutrinos come out with a characteristic energy spectrum, as shown in Figure 11.2
The ovelWhelming majority come from the initial reaction p + P _ d + e + v,_
Unfortunately, they carry relatively low energy, and most detectors are insensitive
in this regime. For that reason, even though the boron-S neutrinos are far less
abundant, most experiments actually work with them.
There are certainly plenty ofneutrinos coming from the sun. John Bahcall, who
was responsible for most ofthe calculations of solar neutrino abundances, liked to
say that 100 billion neutrinos pass through your thumbnail every second; and yet
they are so ethereal that you can look fOlWard to only one or two neutrino-induced
_ .
•
".0
•
.. 10
"
� 10
$ 10
�
r;: 10
V--
".
0
'Be�
0
pep�
·
0
0
'Be�
:�
0
•
----
----
V
-c-:
.,.
�
---,
Neutrino energy (MeV)
Fig. 11.2 The c�tculated energy spectra for solar neutrinos.
(Soura: ). N. Bahcali, A.M. Serenelii, and S. Basu, As1rophYS;·
wi journal 621, L85 (2005).)
"
\\
1189
3"
I
J' N�"lri"" OscillDIWI1S
reactions in your body during your entire lifetime. In 1968. Ray Davis tt at. [2]
reported the first experiments to measure solar neutrinos, using a huge lank of
chlorine (actually, cleaning fluid) in the Homestake mine in SouthDakota (youhave
to do it deep underground to eliminate background from cosmic rays). Chlorine
can absorb a neutrino and convert to argon by the reaction v, + He! -+ 17Ar + t
(essentially again v. + n -+ p + t). The Davis experiment - for which he was finally
awarded Nobel Prize in 2002 - colle<:ted argon atoms for several months (they were
produced at a rate of about one atom every two dill'S)' The total accumulation was
only about a third ofwhat Bahcall predicted [3]. Thus was born the famous solar
neutrino probltm.
11.2
Oscillations
At the time, most physicists assumed the experiments were wrong. After all, Davis
claime<! to have flushed a total of33 argon atoms out ofa tank containing 615 metric
tons oftetrachloroethylene - it was not hard to imagine that he might have missed
a few. On the theory side, Bahcall's calculations require<! an audacious confidence
in the so·nlle<! Standard Solar Model of the interior of the sun. But gradually the
community came to take the solar neutrino problem seriously - espedally when
other experiments, using quite different detection methods, confirmed the deficit.
In 1%8, Bruno Pontecorvo suggested a beautifully simple explanation for the
solar neutrino problem. He proposed that the electron neutrinos produced by the
sun are transformed in flight into a different species (muon neutrinos. say, or
even antineutrinos). to which Davis' experiment was n
i sensitive [4). This is the
mechanism we now call neutrino osciUation. The theory is quite simple - it is
basically the quantum mechanics ofmixed states. which itself is almost identical to
the classical theory ofcoupled oscillators [51. Consider the case ofjust two neutrino
types - say. v, and v". If one can spontaneously convert into the other, it means
that neither is an eigenfunction ofthe Hamiltonian. The true stationary states for
the system are evidently some orthogonal linear combinations:
vl = cosB v,, - sin B v,;
vz = sinB v,, + cosBv.
(ILl)
(Writing the coefficients as sines and cosines s
i just a cute way of enforcing
normalization.)
According to the Schr&iinger equation, these eigenstates have the simple time
dependence ,_iE, ljh:
(11.2)
Suppose, for example, that the particle starts out as an electron neutrino:
v,(O) = l. '1,,(0) =0.
so
vJ(0) = - sinB, II2(0) = cos8
(11.3)
!!.2 OsdlloliollS
In that case
(11.4)
Solving Equaion
t
11.1 for VI"
The probability that the electron neutrino has converted into a muon neutrino,
after a time t, is evidently
(
i
2
2
IV,,(1)l "" (sin8 cos 8) e- E2,/n
=
�
Sl
; (
(
n 28)
sin2 (28)
---
4
1
_
_
e-iEI'/�
)(
)
df2'ln _ dfl'IA
)
d(fl-fll'I" _ e-i(frfll'/h + 1
2 - 2cos
(£2
-
fi
Edt) =
sin2 (28) . 2
4sm
4
---
( E2 - EI )
---,
2ft
(11.6)
You see why they are called neutrino oscillations: v. will convert to vI" and then
ba<:k again, sinusoidally, just as coupled oscillators go back and forth between the
normal modes. In this theory the electron and muon neutrinos themselves do not
have well-defined energies - or masses; the 'mass eigenstates' are VI and V 2, with
masses ml and "'2 .' What is the energy ofa highly relativistic particle of mass '"
and momentum p? Well, £2 - lpl 2cZ _ ",2,4 , so
Evidently then t
,
,
(11.71
• In I"rticular, it is lit�rally noll><,... to sprak ofth� ·rnass· ofan ekctron n�utrino (for �J<amp!e)
- it lias no nuss. an� more dun a three·note chord has a (single) pitch.
t 1 follow here 1M standard derivation. in which p. not E. is held cons�t. Kayser 161 notes dut
this is 'tr<hnically incon�'. but a 'harml�s$ �nor·. sin� it l�.d$ (much more simply) to 1M
right anSWer.
1391
"' I
J J Ntutrirro Oscillations
and hence'
1·
. [1ml - m11" 1 1'
(11.8)
1
' 1'
m
l
[(m - I" 1
(11.9)
t
4hE
P...... .u = sm(20) sm
Write the answer, if you prefer, in terms of the distance z "'" ct the neutrinos have
traveled:
P"''''' Vu =
sin(20) sin
4hE
1
z
In particular. after a distance
L�
"
"
2rr
,,
h::
E",
:
(mi
mt)c)
(ll.lO)
the probability ofconversion hits a maximum, sin 2(20), and at 2L they are all back
to electron neutrinos.
Notice that two things are necessary. in order for neutrino oscillations to occur:
111ere must be mixing (0). and the masses must be unequal - in particular, they
cannot both be zero. It s
i sometimes asserted that the Standard Model requires
that neutrinos be massless, but I don't agree. It's true that some of the calculations
are easier if you make this assumption. but there is no jUndamt:ntal reason why
neutrinos should have zero mass (whereas for the photon this is absolutelyessential).
Cross-generation mixing is a more significant change, though it already happens
n
i the quark sector, and in a way it would be more surprising ifit did not occur for
the leptons.t
11.3
Confirmation
In
2001, the Super-Kamiokande collaboration presented its results on solar
neutrinos [9J. Unlike the Davis experiment, SuperK uses water as the detector
(Figure 11.3), and it is sensitive to muon and tau neutrinos as well as electron
neutrinos. The process is elastic neutrino-electron scattering: v + e --. v + e; the
outgoing electron s
i detected by the Cerenkov radiation it emits in water. The
neutrino can be of any type, but the detection efficiency is 6.S times greater
• Eucdy th� $arne formalism appU... to n�u·
Ira] kaon mixing (Mction •.8.1) � Stt Pro!>·
km 11.2.
t For neutrinos p<.ssing through maIler (as 0p­
posed to vacuum) there are additio�] �ff�ts.
du� to eLostic scattering of �lectron n�utrinos
(v, + . -+ v, + •. by exchan� of a W) and the
zl·mediated inteucrion of neutrinos of any
flavor with �. p. and M. This possibility. first
noted by Wolfenstein. Mikheyev. and Smirnov
171 (he� known origiruilly as the MSW ef.
fect), does not alter the functional form of
Eq....tion 11.9, but it does modify the effective
mixing angle and nuss splitting in a nunner
that depends on � density of the matter and
the tnergy of the beam 18].
11.3 Confirmotjon
Fig. 11.3 The Super-lUImiokande detector (note the p<!Ople in the rubber raft).
for electron neutrinos than for the other two kinds: They recorded 45% of the
predicted number . . . assuming all of these neutrinos were still electron neutrinos.
But remember that their detector is less efficient in counting J.t and T neutrinos. If
some ofthe v;s had converted to v,:s or v;s, then the actual flU){ would be higher
- but how much higher they could not say, because they had no way of knowing
what fracticl1 of the neutrinos had n
i fact converted. You could look back at the
Homestake data (remember, Homestake counted only dutrol1 neutrinos), but the
conditions were sufficiently different that the comparison was not persuasive.
Meanwhile, at the Sudbury Neutrino ObselVatory (SNO) a very similar exper­
iment was under way, using htavy water (D20) nstead
i
of ordinary water. The
virtue of heavy water is that the J1.eutrol1S present admit two other reactions (in
addition to elastic scattering off electrons), and these enable one to measure sep·
arately the electrol1 neutrino fiU){ and the total neutrino fiU){ (Figure 11.4). In the
summer of2oo1 the SNO collaboration published their first results [101, reporting
on the neutrino absorption process (which applies only to electron neutrinos).
They got 35% of the predicted fiU){. If you compare this with the SuperK data
(45%) it appears that 10% of the neutrinos detected at SuperK must, in fact,
have been v" 's or liT'S. But we mow that the detector is 6.5 times more efficient
for electron neutrinos, so f
i they had been v;s, they would have accounted for
• Eta.tic neutrino_eleen-en scattering (.In pr� vi>. Z" el<Change rer aU three ntutrino 'p«ie$,
but for dKtron ntUmnos there i. an extra diagram, mediakd by the W (� Problem ll.l).
1393
3�I 'l
Neutrino Oscillations
Detection Methods
Homestakc cxperimCllt (1968):
v� + 37Cl ..... 3?Ar + e
Super-Kamiokande experiment (19!l8):
Solar neutrino obrervatory (2002);
v+d
v+�
-0
-0
n+p+v
v+e
Fig. 11.4 o.etection rne<:hanisrns at Horn�suke, SuperK, and SNO.
6.5 x 10 - 65%, and 35 + 65 - 100 - right on the money! This was just too perfect
to be an accident, and many people concluded. right then that the solar neutrino
problem was solved, and neutrino oscillations confirmed. Still. not everyone was
convinced, be<:ause this argument n
i volves an awkward concatenation ofdata from
different instruments, taken under different conditions. To nail it down definitively,
the two measurements - the total flux and the electron·neutrino flux - had to be
taken under identical conditions.' Thost results were finally provided by the SNO
collaboration in April 2002 (I2). Suffice it to say that they perfectly confirmed the
tentative conclusions of the previous summer, with
00<01 :::::; Trj6,
(lLll)
(for the conversion of electron neutrinos to muon and/or tau neutrinos).
Ofcourse, the sun is not the only supplier ofneutrinos. There are also terrestrial
sources (radioactive materials, nuclear reactors, and particle accelerators), atmo·
spheric sources (cosmic rays), and astronomical sources (supernovae). In fact, the
first strong evidence for neutrino oscillations was obtained at Kamiokande [13]
(predecessor to SuperK) in the early 1990s using atmospheric neutrinos. Atmo·
spheric neutrinos come mainly from the decay ofpions and muons producedwhen
cosmic rays (high-energy protons from outer space) hit air molecules in the upper
atmosphere:
(11.12)
• Incidentally, you may haw: Men wondering
whelh�r n�utrirlO5 don't simply duay ­
that would cmainly account for the delici!.
But what could th�y decay inUl? t-bybe
somr ev<:n lighter fermion """ never no­
ticed before. This wu actu./.lly a viable (if
implausible) option until the SNO experi.
ments demonstrated c(lnclusively not only
that elKtron ,",utrinos are m><
i ing, but that
the other Aavors are apl""'ring in their place.
Kayser ClDs this the 'smoking gun' evioknce
for neutrino oscillations. It may weU be
true that the hr�virst neutrino, at lust, is
unstable. but its lifetimr is presumably too
long to affect current uperimenl.< [11[.
11.4 Neillrino Mosst"
Evidently, there should be twice as many muon neutrinos (and antineutrinos) as
electron neutrinos.' In fact. however, Kamiolcande found roughly equal numbers
of electron and muon neutrinos. This suggests that the muon neutrinos are
converting to a different flavor. Indeed, the Kamiokande detector was able to sense
the direction from which the neutrinos came; those from directly overhead, which
had traveled only lOkm or so, arrived in the expected ratio (2: 1), but as the zenith
angle increased (and with it the distance to the source), the ratio decreased (see
Problem 11.4). These results were confirmed and improved by SuperK in 1998 (14],
It seems that the muon neutrinos convert into tau neutrinos, with
(11.13)
The atmospheric neutrino experiments (involving muon neutrino oscillations) tell
us nothing about the solar neutrino problem (which n
i volves electron neutrinos),
but it is comforting to see the same phenomenon play out in two different contexts.
The ideal test of neutrino oscillations would involve a fixed source (a reactor
or an accelerator) and a movable detector. As the separation increases, one
would monitor the sinusoidal variation predicted by Equation 11.9. Unfortunately,
neutrino detectors tend to be huge, and oscillation lengths are typically in the
range of hundreds of kilometers (while the flux from a point source falls off like
1/,1). So one must make do with fixed targets and extremely intense sources,
and study the variation with e1le1K}'. The KamLAND experiment [15] uses a new
detector at the SuperK site and looks at neutrinos from several power reactors
IS0-200km away; the MINOS experiment [16] uses a detector n
i a mine in
Soudan, Minnesota, to monitor accelerator-generated neutrinos from Fermilab,
750 km away in Illinois.
11.4
Neutrino Masses
With three neutrinos there are three mass splittings:
(11.14)
Only two of them are n
i dependent (6.31 - 6.21 + 6.n),t The oscillation measure­
ments (Equations 11.11 and 11.13) indicate that one splitting s
i quite small, and
• Of cou�, not all pions dea� to muons, and
Since, bowever, it was alread� established
�l; moreover, kaons as well as pions are
light ntuttinos particil"ting in the weak
not aU muons ckcay ""fOfr rraching ground
produced by (osmic rays. So the factor of two
'' but it sbould be pretty close.
is not '''''1,
t The LSND elCperiment at Los Alamos re­
ported a third ma!;$ splitting incoml"tiblt
with this constraint (17), and was for a wbilt
interpreted as evidence of a fourtb neutrino.
(_ Section 11.9) that the", are a.ttly tbr�
interactions, the 'extra' nrutrino was takrn to
"" 'sterile' (noninteraCling. acept for s.aYity).
At any ratr, thr MiniBooNE experiment at
Fermil.b ltas prttty decisively repudiated tbe
LSND result (18), and with i t tbe notion of
sterile neutrinos.
1 395
3%
I !! Neutrino Oscil!ot!'ons
the others relatively large; we call VI and Vz the closely-spaced pair (with ml > ml),
and V) the loner. This structure is somewhat reminiscent ofthe charged leptons (t
and J.! fairly close in mass, r much higher), and the quarks (d and s close, b higher;
u and c relatively close, t much higher), so it is natural to assume that VI is htavier
than the other two - but it is possible that the neutrino spe<trum is 'inverted', with
V) much lighter than VI and Vl (Figure 11.5).
Unfortunatdy, oscillations are only sensitive to difftnnces in (the squares 01)
neutrino masses, and one would like to measure the individual neutrino masses
directly. This is not easy [19]. The standard method is to study the high energy
cut-off (analogous to 9.2) in the beta-clecay spectrum of tritium, but while these
experiments set upper bounds on the neutrino mass, no measurement to date has
established an actual mass. Meanwhile, an independent upperbound was proVided
serendipitously by the supernova SN1987A: 19 neutrinos, with a range of energies,
were detected in the burst, which lasted only 10 seconds. For massive particles the
speed is (of course) a function of energy, and the fact that these arrived so close
together puIS a limit of about 20eV/,z on the neutrino mass (see Problem 11.5).
On the other hand, the atmospheric neutrino oscillations (Equation 1 U3) imply
that at least one of the neutrino masses must exceed 0.04 eV,e2 . From all available
evidence the best we can say today (200s) is that the heaviest neutrino mass lies
somewhere between 0.04 eV/ez and 0.4 eV/e2..
" ==
'.
" ---
Normal
Inverted
Fig. 11.5 'Norm�I' �nd 'inverted' nelJtrino maH spect'lJm. The lJnits �.e (eVI!?)l.
• Alone amon8 the qmrks and leptons, neulli­
nos could con�iv�bly be their own anti!",rti_
trons witb annihi\;ltion of the accomp.tnying
neutrinos. This should be possible if;;-, .
des - 'Maioram' as opposed to 'Dirac' neu­
trinos (Problem 7.51). [n S«tion 1.5 I Jnen­
�,. but it has never �n obseJVed. One rea·
which appears to demonstrate that v, is dis­
'See·Saw· mechanism. which accounts fOT
the el<traordinary smaUn� of the neutrino
nu.sses by posrulatin8 that they are paired
witb extrem�ly hea"Y neutrinos in a scheme
tioned the Davis and Har"""r elrpefiment.
tinct from iI,. But it could be the hdiciry of
the (anti)neutrino that forbids Equation 1.13.
The ultimate lest is ...ulrillolm double bt/a b;_
""y. in wbich a nucleus with atomic number
Z goes to � nucleus ofatomic number Z +
2. with the emission of two electrons and II<l
....ari"", - in effect. the d�y of two n�u-
son for n
i t�rest in this scenario is that Majo­
rana neutrinos are required by the so-a.Ued
whereby their masses are inversely propor_
tional 1201_ [n any event neutrino flavor osci
l·
lations work the sa,"" for Dirac and Maiorana
neutrinos.
11.5 rheMixi"g MaPi�
11.5
The Mixing Matrix
In Section 1l.2 I discussed oscillations between two neutrino species (v. and vI" for
the sake ofargument). Of (ourse, there are actually three kinds, and this complicates
the algebra a bit.' But the essential point is unchanged: neutrinos inttract as flavor
,igenstaUs (v. is the particle that goes with the electron, vI' with the muon, and v,
with the tau), but they propagate as eigenstates of the free-particle Hamiltonian ­
the mass 'ig,"sta� VI, V2, and VI. lbe flavor eigenstates evolve in a complicated,
oscillatory manner. because really they (any three different masses that are playing
offagainst each other, like the beats ofa coupled oscillator.
lbe same mixing happens with quarks, except that for them the familiar flavors
(d, s, and b) em the mass eigenstates, and it is the 'weak eigenstates' (d', 1,
and b', Equation 9.85) that are 'rotated'; they are the ones that correspond to
the neutrinos.t lbe CKM matrix (Equation 9.86) relates the weak eigenstates to
the mass eigenstates in the quark sector; the analogous construct for leptons is
sometimes called the 'MNS matrix' [22]:
U.,
U"
U"
U.,
U"l
Ufl
)( )
"
V2
VJ
(11.15)
As before (Equation 9.87). it can be expressed in terms of three angles (91l, 6n,
Bll) and one phase factor (,5):
sue
S2JCll
ellCn"
SUCIl
"l
CnC2l - 5ns1l511 t
-C12523 - SI2CBSll eil
-
)
(ll.1G)
(c� = cos B�, sij s sin 6�). But whereas the mixing angles for quarks are all rather
small (so the CKM matrix is not far from diagonal. and the cfOss-generational
couplings are suppressed), two ofthe leptonic mixing angles (611 ::::: 6... and 6n :::::
6Mm) are large. Experimentally, B<l>I 34 ± 2° and 6�,... 45 ± gO. On the other
hand, 61l is known (23] to be less than 10°.
_
• As il rums out. f
i on� of th� tlu� massrs is
subsuntiaUy differ�nl from the others (which
is in fact th��. as � have �n). then
'quas,i·!WO·netUrino oscination' (describnl
by Equation 11.9) remains an excellent
approximation [211.
t There is nothing dap here. Quarks interaa
dominantly bjl the strong interactions, which
are agnostic _ you could use tither set of
_
states: for them ;t is natural to let fI�.or co­
incid� with mass. But nwtrinos only int�r�ct
weakly. so for them it �ms more natural
to use th� weak eigensute5 10 define flavor.
In r�trospect. it would be �ef to spuk
uniformly of 'mass eigensutes' and '�ak
�igenstate5'; tJv, stanohrd Ravors coincide
with mass eigenSla!� for quarks, but with
weak eigenst�te5 for leptons.
1 397
398 1
"
N�(Jlril1o Oscilioliol1S
_
II
,,,
,,,·!:
, :J
; , . .j
Fig. 11.6 flavor cont�nts of th� n�utrino mass �ig�nstat�s.
Btack is v gray is v". white is v" (The eledron-neutrino
compOMnt of Vl is too small to show on this 5nl�).
•.
Because U is a unitary matrix (U-l • UtI. it is easy to invert Equation ILlS,
expressing the mass eigenstates in terms ofthe flavor states:
(11.17)
It appears that Vl s
i an almost perfect
so-so
blend of V" and v, (with a tiny
admixture of v.); V2 is a roughly equal combination of all three flavors; and VI
is mostly v, (Figure 11.6). But it will be several years before we have accurate
numbers for the elements of the MNS matrix, and who knows how long before we
can actually w!cula/e them.
References
1 For a collection of the major papers
on the solar neun:ino problem. see
Bahcall, J. N. (ed) �t aI. (2002) So·
Itzr Neutrin"" The First Thirty Yeors,
Westview, Boulder, CO. For useful reviews of neutrino oscillations.
�e (a) Bilendy, S. M. and P�tcov,
S. T. (1987) Reviews Modem PhYSICS.
59, 671: (b) Haxton, W. C. and
Holstein. B. R. (2000) American jour.
naI of Phy�., 68, 15; (c) McDonald,
A. B.. Klein. J. R. and Wark. D. L
(April 2003) Sci�nlifiG Americon,
40: (d) Waltham. C. (2004) Amer·
ican journol of Physi<s, n. 742.
2 Davis, R. Jr.. Harmer. D. S. and
Hoffman, K. C. (1968) Ph}'$i·
col Review leiters, 20. 1205.
1 Bahal!, J. N., Sahcall, N. A.
and Sluviv. G. {t968} Ph}'$i.
cal Review Laws, 20, 1209.
4 Pontecorvo first discus�d the pos.
sibility of neutrino oscillations - by
analogy with J<?/Ji!' oscillations - in
1957: Pontecorvo, B. (1958) SOM
PhysksjETP, 6, 429. He revived
the idea as a possibl� resolution to
the solar ncun:ino problem in 1968;
(a) Gribov, V. N. and Pontecorvo.
B. M. (1969) Ph�ics Ltttm B, 28,
493.
5 for � ch�rming discussion. seel.yons.
L (June 1999) CERN Cournr. p.
32. For the rel;ativistic tre�tment
�e (a) Sassaroli, E. {1999) Ameri·
con journlll of Ph�ics. 67. 869.
6 for accessible. authoritative. and
well·written material on neutrino
oscill�tions, see pr�ctically mything
by Kayser, S. (2004) This Quote is
From his Ltctun. �t the SLAC Sum·
m�r Institute: which n
i cludes a very
careful treatm�nt of the kin�matics.
See �lso (a) Burkhardt. H. et at
(2003) Ph)l5ics LeUm B. 566. 137.
7 Wolfenstein. L {1978) Physical R�·
view, 017, 2369: (a) Mikheyev,
S. and Smimov, A. (1986) Soviet JOIIrn(11 of Ni<deor Ph}'$ics. 42.
913: (1986)jETP, 64, 4: and
(1986) Nuovo Cimento. 9C. 17.
8 SeeKayser. B..Burkhardt. H. et at
(2003) Physk� Leutrs B, 566, 137.
9 fukud�, S. et aL (lO01) Ph�i·
co:l Review leiters. 86, 5651.
10 Ahmad, Q. R. et aI. (2001) Ph�­
iud Revi= Letters, 87, 071301.
11 hrger, V. eta!. (1999) PhysiC3
Ltturs B, 462, 109; (�) Beacom,
j, F. and Bf,ll, N, F. (2002) Ph¥s·
icaI Rnri..., D, 65, 113009.
12 Ahmad, Q. R. et at (ZOO2) Physi<;aI
Revi..., Ltturs, 89. 011301. For useful
commentary see (�) Schwanschild,
B. (July 2002) Physics Tod(1Y. 13.
i3 Hirata, K. S. et at (1992) Physics LeI·
urs B. 280. 146.
h Eguchi, K. et �l. (2003) Physical Re­
view Ltturs, 90, 021802; (a) Keams,
E., Kajita, T. and Totsuka, Y. (Au·
gust 1m) Scien.tifii: AmericaI', 68.
IS Fukuda, Y. et aI. (1998) Physi·
cal R..,j..., Laters, 81, 1562. For
an illuminating discussion of
the SuperKamiokande atmo·
spheric neutrino experiments
s�e (a) Schwarzschild, B. (Au·
gust 1998) Physics Today, 17.
i6 Michael. D. G. et a\. (20I)6) Phys·
ical R"';..., LttUrs, 97, 191801.
17 Athanassopoulos, C. et al. (1995)
Physical R"';..., Lelurs, 75, 2650,
18 See Schwanschild, B. (June2007)
Physic. Today, 18. For delightful
commentary see (a) Cole, K. C. (June
2, 2(03), Th. New Yorl;u, p. 48.
19 For a survey ofdirect neutrino mass
measurements see Haxton, W. C.
and Holstein, B. R. (2000) Am.,·
ie(111 jount(11 of Physics, 68, IS.
11.5
Tht MixingMatn.. 1 3fl
20 For interesting commentary on
the distinction between neutri·
nos and antineutrinos seeBoas, M.
(1994) A"",ricall joum(11 of Physics,
62. 972; (a) Hammond, R. (1995)
Am.,ical' journal of Physics, 63,
489; (b)Wagner, R. G. (1997) Amer·
ican joum(1! of PhysiCl, 65. 105;
(e) Fewell, M, P. (1998) A"",rj·
call journal of Physics, 66, 751;
(d) Holstein, B. R. (1998) Amer.
ical' journol of Physics, 66, 1045;
(e) Fewell. M. P. {1998} Ameri·
call JournaL of Physics, 66, 751; (f)
Boya, L J. (2000) American jour_
....1 ofPhysics, 68, 193. On neutri·
noless double beta decay (,,BJI(Ov))
see (g) Elliott, s, R. and Vogel, P.
(2002) Allnual R.IIi..., of Nudtar
and Partick SciellCt, 52, 115.
21 See Kayser, B., in R.vi..., of P(1r·
tidt Physics (2006) p. 156.
22 This is in honor ofMaki, Z.,
Nakagawa. M. and Sakata, S. (1962)
whose pioneering work Progress in
Throretical Physics, 28, 870, long
predates the discovery of neu·
trino oscill�tions (or of the tau),
23 For planned experiments to mea·
ber 201)6) Physics Today, 31.
sure ell. see Feder, T. (Novem·
Problems
il.1 Estimate the lifetime ofthe sun, assuming (as Lord Kelvin did) that the source of the
energy radiated is gravity. Look up any empiricOll nwnbers (the power radiated by the
sun, the mass, and "adius ofthe sun).
(.) What is theperiodof1(0 "'" It oscillations (Section4.4.3)� IHilll:Themasseigenstates
ar� K1 and Kt. In the neutrino case (Eq. ll.7) the particles were highly relativistic;
for the K's, assume on the contrary that the kinetic energy is substantiOllly 1m than
the rest energy.]
(bt Compareyourresult in (a) to the lifetimes ofK1 and Kt. Noticeth.at the � component
of th� beam dies out - le�ving pur� Kt - well before signific�nt oscillation can
occur.
i1.3 Draw the lowest-<:>rder diagrams for elastic neutrino-electron scattering, (a) fordectron
neutrinos, (b) for muon neutrinos, (e) for tau neutrinos.
11.4 (a) Suppose atmospheric neutrinos are produced at an altitude h, and the detector is at
sea l<'"Vel, Find the distance x from the source to the detector, as � function ofth� zenith
angle e (dire-ctly overhe�d is e _ O. the horizon s
i e _ 9Qo. straight down is e _ 180").
Let R be the radius ofthe earth.
il.l
<100
I
J!
Neulrif10 Oscillations
(h) Suppose 95% ofthe 'upper' neutrinos (overhe�d to horizon) re�ch the det�tor, but
only SO% ofthe 'lower' neutrinos (bdow !he horizon) do. Using the oscillation formul�
Equation 11.9 (but this time for muon neutrinos converting to tau neutrinos), determine
6 and 1l.1TI2. Assume h _ lOkm and E _ IGeV. [This problem was posed by W�lth;om II].
You'lI need � computer to get the numeric�l �nswer.]
1l.S (a) Show that the velocity of an ultra·relativistic �rtick (mass ITI) with energy E is
approximately
(h) Supernova SNl987A occurred in the urge Magellanic Cloud 11.7 x lOS light years
from Earth). Neutrinos from this explosion, with energies ranging from 20 MeV to
30 MeV. were det�tedwithin a 10 s time interv.r.L What upper bound on the neutrino
mass does this imply' (Assume the neutrinos all started out at the same instant]
11.6 Wi!h neutrinooscillations. !he individual lepton numbers (I.. L� , �nd L,) are no longer
conse-rved, and this means that the d�ay I-' .... e + y Ithe absence of which suggested
these (ons<:rv�tion I�ws in the 6rst pl�ce - see Eq. L16) is possibl., in principle.
(.) Draw � feynm�n diagram for this process. Note: neutrino oscillations can be
represented by ... blob:
MIn this process you must 'borrow' the energy necesury to make the virtu�1 w.
According to the uncertainty principle (see Problem 1.2), how soon must you 'repay'
the debt) How rar could a neutrino get in this time? Given that neutrino oscillations
OCCur OVer dislilnce scales of many kiloml:u", dOts it seem likely that you could
'borrow' the energy long enough for I-' --+ • + y 10 occur?
"
I'
12
Afterword: What's Next?
So for. I have talked almost exclusively about establishtd 'facts'. With tk possible
exctptwn of the Higgs muhanism, any Juture theory will havt: to indwk aU of this.
But tm Standard Modd is certainly not the last word on the subjett. Already there are
intriguing thcorctkal spuulation5 and tantalizing experimental indications ofwhat the
Julurt: may held. [ncreasingly, the impetus is comingfrom obstlV«tions in astrophysics
and cosmology, rather than traditional coUUkr fXperimwts.* In this chapter. I'U txplon
some of the directions in which future discovtri� sum most likdy. I'U start (Staion
12. J) with the huntfor the Higgs, which lopS the agmdafor the Large Hadron Collider
(LHe) (andfor the remaining lifetime ofthe Tevatron), and may lead to the explanation
for aU particle ma=. Next (Section 12.2), 111 discuss Grand Unijication, which was
the 'natural' next sUp 30 year.; ago, but hit a brick wmI wmn the prwic�d decay of
tm proton LWS not obsuwd; it ne�rtm/a;s sets the con�xtfor aU subsrqmnt theontical
de�lopmen/.s. Then (Section 12.3) /'U consider CP violation and its implicatiOns for
the mat�rlantjmatter asymmtJry of the universe. Section 12,4 is a scandalously brief
introduction to supusymmetry, extra dimensions, and sIring theory, ideas that havt
dominated theoretical partic� physics sinu 2984 and for which the first eXFrimental
support may comefrom the LHC. FinaUy, in Section 12.5 we'U study Dark Matter and
Dark Energy, which by curnnt eslima!.ts accountfor 95% ofthe matter in the universe,
ltavingonly a paltry 5%forthe 'ordinary'particleswt' encoun�rt'd in thefirst 1 1 chap�rs.
12.1
The Higgs Boson
[n the Higgs mechanism, a gauge symmetry is spontaneously broken by a
two-component scalar field
whose ground state is not zero (Section 10.9).
One component of is reincarnated as the third (longitudinal) polarization state
tP
tP,
• In r�trospttt, one might caU !h� pniod from
!he early 1930s to 1954 !he er� of cosmic r�ys,
and from the Cosmotron to the large Hadron
Collider (LHC) - let's say 2010 - the era of
accelerator physics; in this sense. we are now
entering the era of particle utrophysics (I).
Part of the reason is simple economics: to
reach �r higher energies. accelerators have
become so hu� and $0 e"l"'nsi"., mat it is
hard to imagine anything beyond the Interna·
tional tinear Collider (ILC) now on the draw­
ing ba.rds. Astrophysics offers a relatively in·
expensi"" window into vastly higher energy
regimes.
Introduction '" E/",,,n
,, t4ry P�rtida, St.:Md Edition. David Griffiths
Copyright C 2008 W1LEY·VCH Vulag GmbH & Co. KGaA. Weinheim
ISBN: 978·J.S2H06O[·2
<1021
11 A�fWOrd; Whul" Nut?
for a now massive gauge field, but the other remains, representing a neutral particle
of spin 0: the Higgs boson [21.
i the Higgs mechanism because it seems to be
Most particle physicists believe n
the only way (certainly it is the clwntst way) to account forthe mass ofthe W and the
Z, in the context oflocal gauge theory. But ifthere really is a Higgs field, permeating
all of space, with a nonzero value even in 'vacuum', it could account as well for
the masses of the quarks and leptons, whose interaction with the primordial Higgs
field has been likened to wading through deep water, imparting an effective inertia
to (almost) everything that moves. In this more exalted vision, the Higgs particle
becomes the source of aU mass." The quarks and leptons are 'born' massless,t
but with Yukawa couplings (Problem 10.21) to the ¢: Lint = -aftfVrf¢' where
f denotes the particular quark or lepton. When ¢ is 'shifted' by spontaneous
i a
symmetry-breaking (Equation 10.130), Lint splits into two pieces, one ofwhich s
Yukawa coupling to the phySical Higgs field and the other a pure Fermion mass
tenn, - mf ,l�J'/IJ (in the notation of Section 10.9, mp - (/.I-/A)afi. Unfortunately,
this doesn't help us to calculate the particle masses - it simply trades one unknown
parameter (m.r) for another (afi. But it does suggest that the strength of the coupling
to the Higgs is proportional to mass.
In the simplest theory (the 'Mn
i ima! Standard Model', MSM), there are four
by the W± and z!J (which thereby acquire mass) and the fourth remains as the
neutral Higgs field. More complicated schemes have been proposed, involving
multiple or composite Higgs partides,* but the MSM provides a useful roadmap
for experimental and theoretical exploration ofthe Higgs sector. In this model, the
Higgs (h) interacts with quarks and leptons by the diagram
scalar fields to begin with - two charged and two neutral. Three ofthese are 'eaten'
I
I
Ih
I
�
(vertex factor -impIv), and with the weak mediators by
,
I
Ih
I
,
I
Ih
,
� �
t In th� Standard Model Lagrangian, fermion mass terll1$ /i"t) are nOl n
i variant under the eltc·
• ILon Lfflennan r:unously ulled it '1M. God P"nid<· (N�w York: o.:ba, 1993).
trow�ll< symmetry SU(2), )( U(I). SO the ·starting· nusses of the quarks md leptons have to be
zero, and th� physiul mas�s arise only when rru, symmetry is (spontaneously) broken.
* In $uptrsymme!ric th",ries. for elYlmple. there are at least five HiSS" bosom and in technicolor
the role of the Higgs is piayed by a bound sta� of two fermiolU.
11.1 Tht Hjggs80son
(vertex factor 2iM�rg"" /(n2v), where the subscript m stands for W or Z). There
is, as well, a direct Higgs-Higgs coupling'
,
1
Ih
1
/
...
... ... h
A,
....
h .... ...
.. ,
,
p
{vertex factor -3im�c2/(n2v)). Here, v is the 'vacuum expectation value' of </>1 (p
for the potential in Section 10.8). It can be calculated from the mass of the W (see
Problem 12.1)
(12, 1)
�
2Mwc2
"lie v = -- = 246 GeV
gw
The mass oflhe Higgs itselfis not determined by the theory.t
It would be nice to know whether this story (or some variation on it) is actually
true. The Higgs particle is the only element in the Standard Model for which there is
as yet no compelling experimental evidence. [t may have been seen al LEP (CERN),
in the lasl months before it shut down (to make way for the LHC) 14), it could
slill be found al lhe Tevatron (Fermilab), and unless current thl'Qries are wildly off
it will certainly be observed al the LHC. Various conslraints - experimental and
theoretical - suggest that its mass must lie in the range
114 GeV/r < II1/, < 250 GeV/e2
(12.2)
with a most probable value around 120 GeV/e2 [5), The LHC will explore the entire
region up to 1 TeV and beyond.
At LEP (an electron-positron coUider) the Higgs was sought in the Z·'bremss­
trahlung' reaction e+ + e- --+ Z + h:
,
,
z
z
At hadron colliders (the Tevatron and LHC), the dominant production mechanism
is gluon 'fusion', g + g --+ h via a quark loop (mainly the top, since it's the heaviest,
• n..,r� ar� also 'four·poin!' couplings Ilk ..... ZZ, kll ..... ww, and 1m ..... kk (l).
t In Equation 10.121, m. invo]""s only /-', not "lA, SO il is sensiti"" to th� shaFt of th� I"l",ntial,
not ju" the Vaouum �pectalio" value of 41,.
1 0403
04j 12 !ifurword:
Wmu's Nt.:!?
and hence has the strongest coupling to the Higgs):
but several other modes are expected to contribute, notably W/Z-bremsstrahlung:
q'
q
W{Z)
W{Z)
and W/Z fusion:
q
h
q
(direct quark fusion, q + q -+ h, does not contribute much in the MSM. because
the only readily available quarks are u's and d's, which, since they are very light.
couple weakly to the h).
How do we expect the Higgs to decay? Because Higgs couplings are proportional
to mass (or mass squared, in the case ofthe W and Z), heavy daughters are favored.
if they are kinematically allowed. The branching ratios depend a lot on the mass
of the Higgs (see Figure l2.1). If m� is less than about 140 GeV/cl, the dominant
mode is h -+ bb, but above that h -+ W+W- takes over (with a virtual W up to
160 GeV/c1 and real W's from then on); h -+ ZZ is close behind (especially above
180 GeV/c1). and n
i the unlikely event that the Higgs is heavy enough to make two
tops (� > 360 GeV/c2) h -+ Ii assumes third place. More exotic decays are also
possible, such as a photon or gluon pair:
<
h�
___
_
,
_
0
�
These decay rates have all been calculated in great detail (6J (you can do some of
them yourself - see Problems 12.2 and 12.3).
11.1 Grund UnifiWlion
..
1.0
0.1
0.01
. .
:g�!\
,
f
f
,
\.
100
/
.
.
' .
.. .. .
................ _-
.
ww
i
!
j
'./
I
,
,
\
i rr :>:� i
: ......y. \, t
"
..
j
cr
,,
f
.. .
.
......
/"o----------------Zi--------------------··.·.··--···---.......
" S_ "
.
. . ..
,
,
,
•
h
.... \\
bb
200
300
In
f
//-_.•..
/
/
Fig. 12.1 8rilnching ratios for Higgs d�cay, as functions
of th� Higgs maSS (in GeV{d). (Source: Gunion, ). F. eu.1.
(1990) The HiW H�ntu'5 Gl1ide, Addison-Wesley, Redwood
400
500
Qty. CA.)
As soon as the Higgs mass is established, one will be able to draw a vertical
line at the appropriate point in Figure 12.1 and read off the branching ratios. If
the measurements disagree (as they probably will), then the Higgs sector is more
interesting than the MSM contemplates. And of course if no Higgs particle is
found at all, then we have a revolution on our hands.
12.2
Crand Unification
With the success of electroweak unification, in the 1%05, the logical next step
was to n
i clude the strong interactions, in a 'Grand Unified Theory' (GUT) that
would identify aU three forces as different manifestations of a single underlying
interaction. Of course, the strong forces are enormously more powerful than the
others; but the same could be said of electromagnetic versus weak forces, and
we now understand that disparity as an artifact of the huge mass of the W and
Z - their intrinsic strengths are quite similar, but it is only at energies well above
MII/C2 that the unity becomes manifest.
Moreover, as we saw in Sections 7.9 and 8.6, the coupling 'constants' themselves
are functions of energy - the strong and weak couplings go down, while the
electromagnetic coupling goes up. It s
i r
i resistible to suppose that they coalesce at
some point (Figure 12.2); above the grand unijication scale (:0:::1016 GeV) there is just
1 405
"' I 12 Aftuword: Wha!', Nex:?
80
.- '
60
"
20
0
-,
.,
('I
80
60
.-,
40
-,
.,
'
10
20
'
10
lO
in
"
10
"
10
(bl
-,
.,
-,
.,
a)""
'
10
'
10
'0
10
"
10
ro"
Fig. 12.2 Conv�rg�rlce of the coupling constants at the GUT
scale (al in the Minimal Standard Model. (b) with super·
symmelI)'. The horizontal axis is energy. in GeV.
one universal coupling constant, and the strong. electromagnetic. and weak forces
are identical in strength:
The first (and simplest) GUT was introduced by Georgi and Glashow in 1974
[7[. It led to a spectacular prediction: the proton is unstable. decaying (for example)
into a positron and a pion
(12.3)
The lifetime is reassuringly long - at least lOW years, which is loW times the
age of the universe - though (since we have easy access to a lot of protons) not
beyond the range of measurement. [n 30 years of n
i creasingly precise experiments,
however, proton decay has never been observed [8]. The current lower bound is
Tproton >
lOll years
(12.4)
(which probably vetoes the Georgi-Glashow model). More elaborate GUTs have
been proposed, but almost all of them require proton decay at some level.
Although there is no direct experimental evidence in support ofgrand unification,
beliefin it is an uncontested article offaith among theorists. In a way, the 'natural'
evolution ofparticle physics was rodely interropted by the failure to detect proton
decay. Had proton decay been discovered in - say - 1985, one can easily imagine
that enormous efforts, theoretical and experimental, would have been devoted to
Aeshing out the details of grand unification, just as the previous two decades
had Aeshed out the Standard Model. But that's not what happene<l, and today
grand unification simmers, half<ooked, on a back burner.t What are its essential
features, and why should we take it seriously? [9].
• Awkw�rdly. it is now de�r thol they do no! (quite) meet at � sin� point, in t� MSM; one of
t Tesl�ble predictions of gr�nd unification. �I Ihe relatively low energies presently acceuible, are
the attr�clions of sUf"!'rsymmetty is that it makes perfect con�gence possible.
few and f�r between. Proton decay. if it exists. is the best probe avail/;ble. but we are fasl ar­
prOilching the practical Limit on proton lifetm
i e measurements (see Problem 12.4).
12.2 Gralld UllifiCOlioll
Table 12.1 Fermion Stales in the SU(S) GUT
Quin�t
Decuplet
Grand unification contemplates an overarching symmetry group (SU(5), in the
Glashow-Georgi version) that contains as subgroups the (color) SU(3) and S U(2)L
o U(I) symmetries of the Standard Model. The fundamental fennions (quarks
and leptons) are assigned to representations of this group. much as the Eightfo(d
Way assigned baryons and mesons to (octet, nonet, and decuplet) representations
of (flavor) SU(3). The first generation comprises 15 particle states: u and d, each
in three colors and two chiralities (L and R), e (L and R), and v, (L only '). [n the
SU(5) GUT, they constitute a quintet and a decup\ett (Table 12.1); in the absence of
symmetry.breaking (presumably by the Higgs mechanism), the states in each mul·
tiplet share the same mass and interact identically. (The same goes, ofcourse, for the
other two generations.) There are 24 mediators.\: (Table 12.2): the 8 gluons, the pho·
ton, W+, W- , andZ, and I2newones - the X (charge ±4/3, 3 colors, hence 6in all)
and the Y (charge ±1/3, 3 colors, for another 6). Theycoupleleptonsto (anti)quarks,i
and hence are known as /tptoquarks. For n
i stance, d -+ e + X and u ....... e + Y:
y·1
� �
X.!
Table 12.2 Gauge Bosons in the SU(S) GUT
Chilrge
MilSs
8 gluons
0
0
1 ph oton
0
1 W± , Z
L -1,0
-1(1 GeVf'z
6X
6Y
Ifl. -1/]
4/1, -4f]
0
_10'6 GeVfcZ
_1016 GeV/cl
• In 1974, it was assumed that the neutrino is massless, and lhe fact t�1 there wu no mtural
p\.oce for "_ WaS token as a virtue ofthe theory. For massi� neulrirlO$ "_ must be assi�,
awkwardly, to a singld repr�ntation of SU(5f - or, in the case of Majorana neutrinos. the
Higgs sector must be expanded.
t The fact that they don't all fit i nto a sinsle r
i reducible representllion is an unattracti� feature
of the SU(5f modd: the SO(lOf GUT usigns all tS, plus III, to a llKl.imensional repreSl!ntotion.
.j: In �neral, SU(IIf has III - 1 mediators (eisht gloons for color SUm, thr� intermedia� vector
boson. for SU(2fLl: U(nf has n (hence One photonf.
i Notice that il is the anli-d that li6 n
i the same multiplet u the electron.
1 407
408 1 '2 Aftuword; Whot's NeX!?
•
•
,
"
"
:.---:;-----.: (itO) (p)
,
•
'pi
Fig. 12.3 Prolon dea1 in the SU(5) GUT.
(p)
They also couple quarks to antiquarks (in this context. they are sometimes called
diquarks). as in u -+ Ii + X and d ..... Ii + Y:"
y>l
� �
x>1
This larger symmetry is badly broken, obviously (quarks and leptons do not have
the same mass, and the strong interactions are - well - stronger than the others).
Just as eJe<:troweak symmetry becomes apparent at energies well above the Wj2
mass, GUT symmetryprevails at energies above the (huge) grand unification scale.
That's why it is so difficult to test grand unification in the laboratory - even though
its implications are, in principle, dramatic. The leptoquark couplings allow for
nonconsetvation of lepton and baryon number, and hence license the decay of the
proton, via diagrams such those n
i Figure (12.3). But because these mediators are
so heavy (presumably in the neighborhood of the GUT scale: Mx ,..., My '"- 1016
GeVfeZ), the decay rate is extremely small (Problem 12.5).
Apart from the largely aesthetic attraction of unifying the fundamental forces
of particle physics, grand unification purports to 'explain' the relation between
quark and lepton charges (and beyond that the quantization of charge itself).
For technical reasons the sum of the charges in a multiplet must be zero, and
putting quarks and leptons into the same multiplet forces (in the case ofthe SU(5)
quintet)
(12.5)
• Ostensibly these re�ctions do not conse� color, but remember that the 'cross product" of \W<)
color states carries a single color (Equation 10.84), and it is such a (ombi(1.l.tion thaI is implied
here
12.3 MOlu'/Atllimoller Asymmetry
Our world would be a radically different place if the electron and proton did not
have precisely opposite charge, but short of grand unification there is no reason of
principle why this had to be so:
12.3
Matter/Antimatter Asymmetry
Everyone assumes that the Big Bang created matter and antimatter n
i exactly equal
amounts. If this s
i the case, how come we are surrounded by electrons, protons,
and neutrons, with no positrons, antiprotons, or antineutrons in sight? Of course,
if a positron (for example) dots show its face, it doesn't last long: as soon as it
encounters an electron, they annihilate. But this doesn't explain the preponderance
of leftover electrons. Perhaps it's a local phenomenon - our matter-dominated
corner of the universe is balanced by an antimatter region somewhere out there.
However, there is no evidence for this - on the contrary, astrophysical observations
indicate that the known universe, at least, is all matter (if there wtrt an antimatter
zone, the border would be an extremely violent place, and it is hard to imagine that
the cosmic microwave background would show no sign of the disturbance) [12].
Alternatively, some process must have favored matter over antimatter in the course
ofcosmic evolution. What sort of mechanism might do the job?
In 1967, Sakharov [13] identified the necessary ingredients. Obviously, there
must be an n
i teraction that violates conservation of baryon and lepton number
(something grand unification could supply). There must have been a period when
the universe was substantially out of thermal equilibrium (otherwise any reaction
i ...... f would go just as often the other way,f ...... i, and there would be no net change
in baryon number). And, crucially, there must be CP violation - some reaction
i ...... f whose rate is different from its CP-conjugate, 7 ...... J (otherwise, again, there
would be no net change n
i baryon number). Conveniently, CP violation had recently
been discovered by Cronin and Fitch in the �/I(!! system.
To this day, the underlying nature ofCP violation is not well understood. Parity
violation was very easy to incorporate into the theory of weak n
i teractions: one
simply replaced vector couplings, yl' , by vector/axial vector couplings, yl' (1 yS)
(Section 9.1). But the only known source of CP violation is the residual phase
J in the CKM matrix (Equation 9.87), and it s
i hardly obvious why this breaks
CP invariance. Consider a process i ...... f, and the CP-reversed process 7 ...... (for
instance. if i includes a left-handed electron. i includes a right-handed positron);
CP violation means that the rate for i ...... J is not the same as for i ...... f (for
_
J
• A mor� probl�m�tic implic�tion of grlnd
unificllion i5 Ih� �is�ru:� of sUp<'!" huY}' 'I
Hooft-Polylkov INgnetic monopole5 (10).
which should be pr��nt n
i l.u� num!>'r.
(left over from the Big Blng). but hlve never
bttn okt�ted in th� bboralory (weU .
mlY!>. ana (11)). [nHltion�ry cosmology can
�((aunt for � dilution in th� num!>.r. but
the prediction of (unoh!;erved) monopol�
in grand unifi",tion - and for that mltter in
other th�ri� �$ well - r�lruIins � troubling
problem.
J 409
410
I
12 AftelWOrrI: Wh",'s Nut?
V.. <:
,
_
_
_
b
v.,
"
(IC')
w
"
,
(K�
"
::::.---:---- (:n:-)
b
(BO)
v"
V"
Fig. 12.4 Two diagrams for ffJ ...... IC' + If - . The second is a 'penguin' (Problem 4.40).
example, If! ..... K+ + ]'(- is 13% more common than If ..... K- + ]'( + ), Now, the
amplitude, A, is a complex number, and ordinarily it is the same for i ....... as for
i ..... f - =pt that any CKM element gets conjugated. Thus
j
(12.6)
i the 'conjugating' phase and ¢ the 'ordinary' phase." On the other hand,
where 9 s
reaction rates are proportional to )AI2, so there is no CP violation, even though
the amplitudes themselves are different.
But suppose that the process (i ..... fJ can proceed by two diffntnt routes (for
example, If! can go to K+ + ]'(- in several distinct ways - see Figure 12.4), Then
A - Ji1 + A2, with
(12.7)
(12.8)
It follows (Problem 12.6) that
(12.9)
In this case the rates are not the same, and CP is violated. Notice that there has
to be a conjugating phase (from the CKM matrix) as well as a nonconjugating
phase - and these have to be dijftnnt for the two contributing routes.
From the experiments, we know that CP violation occurs in the weak interactions
ofquarks, and is attributable to the phase factor in the CKM matrix.t Unfortunately,
this is nowhen nt:ar enough to accountfor the matter dominance ofthe universe [IS],
• In th� Iit�ratur�, thq a", so�im� (anr<! '_a\:' and '.trong pm..,., respe<tivdy. Th� distill('
tion is subtle, but in practic� e (omes exclusively from the CKM mmix element, and � typically
involves strong interaction efT"Cls (14).
t In f;oct, all such CP violatins elf...,ts ar� proportional to th� height of tm, 'unitarily triangle'
(Problem 9.B).
12.4 SLlP<"rsymmelry. S'ri�� Extra Dimensions
so one is forced to speculate aoout other mechanisms of CP violation. With
massive neutrinos and the leptonic analog to the CKM matrix (Section 11.5).
the same phenomenon should occur in the lepton sector. where it would reveal
itself. for example, in W1equal probabilities for v. -+ v" and ii, -+ Ii.... This has
not been observed (yet). but it s
i conceivable that it would provide a mechanism
(sometimes called /eplogentsis'") for the observed matter/antimatter asymmetry.
Another possibility is CP violation in the strong n
i teractions (in this case the
'smoking gun' would be a nonzero electric dipole moment for the neutron). CP
violation has never been observed in strong processes. but there does not seem to
be any fundamental theoretical prohibition} At this point, the matter/antimatter
asymmetry of the universe remains an uncompleted puzzle; the essential missing
piece is the nature of the CP violation responsible. It is far from clear how this
story will resolve itself.
12.4
Supersymmetry, Strings, Extra Dimensions
12.4.1
Supersymmetry
The classic symmetries ofquantum mechanics involve different states ofthe same
system. Rotational n
i variance, for instance. requires that the theory be W1changed
when the state '" is replaced by its rotated version U(9}'" (Equation 4.27) - or, more
precisely, the Lagrangian is unchanged (in first order)t when the wave function
is incremented by the infinitesimal amount 8", = (-illi)[�9 . 5]", (Equation 4.28).
Particle physics long ago generalized the idea to 'internal symmetries' involving
closely related particles (flavor multiplets. for example). In 1974. Wess and Zumino
[17] introduced a more radical symmetry that stirred together fermions and bosons.
For example. a scalar field ", could mix with a spinor field '"
(12.10)
where ( is an infinitesimal spinor describing the transformation (analogous to 89
for rotations). and ? ... (tyO is its adjoint. What if we insist that the theory be
• The !ermill<llogy is not entirely consistent baryogtnrn. is the generic word for the origin ofnut·
ter domin�nce. so Iepto�nesis is acrn;illy one possible me<:h�nism for b<.ryo�""sis.
t lnd�. it is something of a mystery why strong CP violation does nol occur. One pos$lble ex­
plan�tion was suggested by Peccei �nd Quinn 116) in 19n, � neutnl spin.Q particle (the =ion)
couples to the qu�rks in such a way u to cancel dynamically any .trong CP violation. Axlon.
have not been observe<!. but they renuin among the viable candidates for dark nuller.
� It i$ generally shnpler to WQrk with infinitesimal transfornutions. and there is no loss of gener.
�lity $lnu: a finite transfornution can be built up as � sequence of infinitesimal ones (Problem
12.7).
1411
"12 1 12 Iojierword: Who" s N�)(I?
invariant under such a transformation? [t s
i not hard to construct a Lagrangian
with this property; the combined free Klein-Gordon and Dirac Lagrangians are
n
i variant
(12_11)
as long as the boson 4> and its fermion partner 1/1 carry the same mass (Problem
12,8). A similar game can be played joining a particle of spin 1/2 to a particle of
spin 1 - and in general pairing particles whose spins differ by 1/2. [nvariance of
this kind, linking fermions and bosons, is called 'supersymmetry',
Over the past 30 years an enormous amount of work has been done on su­
persymmetry [18), and [ think it is fair to say that most particle physicists are
convinced (without as yet any supporting experimental evidence) that it is a fun­
that every fermion has a bosonic partner (identified by putting an 's' in front of the
damental symmetry of nature, Supersymmetry carries the stupendous implication
name - thus 'squark', 'slepton', 'selectron', 'sneutrino', etc.) and every boson has
a fermionic partner (identified by putting an 'ino' after the name - thus 'photino',
'gluino', 'wino', 'higgsino', etc.). Where are all these particles? If supersymmetry
were unbroken, they would share the masses oftheir 'ordinary' twins - the photino
a mass of 0.511 MeVjc!, This is nonsense, obviously - no such particles exist.
would be a massless particle of spin 1/2, and the selectron a spin·O particle with
So the symmetry must be badly broken (perhaps spontaneously, but there are
other possibilities, especially if gravity is brought into the picture). Presumably
the supersymmetric particles are much heavier - too heavy to be produced by any
existing machine, though there are strong indications that at least some of them
should be accessible to the LHe.
Hmm . . . . Why should we take such an outlandish scheme seriously? Super.
symmetry has the potential to solve several thorny problems, among them the
following:
1. By introducing a number of new particles, it modifies the
energy dependence of the three running coupling constants
(see Equations 7,191 and 8.94), making possible their perfect
convergence at the GUT scale (Figure 12.2).
2. [t offers a 'natural' solution to the so-called hierarchy
diagrams (Section 6.3.3), which drive it way out of acceptable
problem. The Higgs mass is renonnalized by various loop
range unless there are magical cancellations ('fine tuning').
But loop corrections are ofopposite sign for bosons and
fennions, so supersymmetry, by pairing particles with
'sparticles', makes the cancellation exact and automatic.
3, [n most models, the lightest supersymmetric particle is
colorless, neutral, and stable, making it an attractive
candidate for Dark Matter (Section 12.5).
� .,
12,. Sl'penymmtlry, String<, &lrll Oim.n5io",
,-------�
--,
"roo,
.
Ele<:trOlTlagl""letlc
Weak
Gravitational
I
1
Gral""ld
Electroweak
.
Theory of
UI""II Icatlon
1
everything
t
Fig. 12.5 Unification of the four interactions.
Moreover, attempts to formulate a quantum theory of gravity seem to require
supersymmetry. On the other hand, the minimal supersymmetric models involve
at least 124 independent parameters (19) - five times the (already embarrassing)
number in the Standard Model - and they do not easily accommodate neutrino
masses. If supersymmetric particles are discovered' at the LHC, it will be a
spectacular triumph of inspired audacity. But I wouldn't bet your last dollar on it.
12.4.2
Strings
For decades, a fundamental challenge in theoretical physics has been the for·
mulation of a quantum theory of gravity - the quantize<! version of General
Relativity (analogous to QED, the quantize<! version of electrodynamics). Gener·
ations of physicists have tried, and failed - for point masses the theory seems
to be incorrigibly nonrenormalizable. While this is embarrassing, it has not, so
far, been catastrophic for particle physics, where gravity is much too feeble to
play a significant role. But at extremely close range (which is to say, at very high
energy - spedfically, the Planck scale: 1019 GeV) quantum gravity is bound to
come into the picture. Moreover, the old dream of unifying the forces of nature
leads inexorably to a putative 'theory ofeverything' that would include gravity along
with the strong, electromagnetic, and weak interactions (Figure 12.5).
String theory proposes to solve these problems (and more) [22]. In string the·
ory the basic units of matter are not (zero.dimensional) particles, but rather
one-dimensional 'strings' (or higher-dimensional 'branes'), of which 'particles'
are various vibrational modes. The theory underwent an extraordinary evolution
between the 1970s, when a few lonely visionaries took up the cause, and 2000, by
which time it was well established as the dominant paradigm. Early versions con­
tained only bosons, and consistency required 25 space dimensions. This seemed
a trifle extravagant, but it was possible to imagine that 22 of them are 'curled up'
•
There was a flurry of excitement in 2001. when discrepancies betwttn thc measured and Qku·
law! values of the ano=lous mag""tic moment of the muOn se.emed 10 suggest a contribution
from supersymmelric p,articles (20). But il rurned OUI that Ihe Qkulatioru; w<:rc in error - when
a sign mis�ke in one term was corrected the disagrttment largely evaporated (21).
1
413
414
] 11
/ifttrword: What's NtxI?
(rompacfijieJ), and hence irrelevant on the macroscopic scale.' Fermions were later
incorporated via supersymmetry (hence 'superstrings') and the number of space
dimensions dropped to 9 or 10. Meanwhile, it was realized that the theory auto·
matically includes the graviton, making it a natural candidate for quantum gravity.
In the early days, one of the great attractions of superstring theory was that it
appeared to be uniquely detennined - we live, it seemed, in the only mathematicaUy
possible world. Physics would no longer be a matter ofdiscovering contingent laws
by experimental observation, but of working out the inescapable implications of
the one allowed theory. Sadly, this particular hope has turned inside out, and
'M-theory' now suggests that there is in fact a whole 'landscape' of permissible
models (10500 of them, by some estimates), and no way (short of the anthropic
principlet) to choose the correct one.
At this point an entire generation of theoretical physicists is way out on a limb.
Superstring theory still holds out the best hope for ultimate unification of aU four
interactions, and it is probably the most promising candidate for quantum gravity.
But it has proved diabolically difficult to extract verifiable (or falsifiable) predictions
about the low-energy world we inhabit. The discovery of supersymmetric particles,
or indications of extra dimensions [23J, would lend some support, but anything
approaching a confirmation of superstring theory seems, at this point, a very long
way off[24J.
12.S
Dark Matter/Dark Energy
Persuasive astronomical evidence now indicates that the matter we know about ­
described by the Standard Model - represents a measly 5% of the mass/energy
content of the universe. The rest is Dark Malter (about 20%) and D«rk Energy
(75%). The implicaions
t
for particle physics are humbling: we have only seen the
tip of the iceberg. What is aU this other stuff, and how has it managed to elude
m'
12.5.1
Dark Matter
In 1933, Fritz Zwicky measured the velocities ofgalaxies in the Coma cluster (from
the Doppler shift oftheir atomic spectra), and use<! this information to determine
• � id�a of extra dimensions was nOI 1l'"W. T.
K.aluz.;r first introduced 1M notion in 1919, in
an �(fort 1(1 unify �1'"Ctrody�mics and grav.
ity. and in 1926 O. K!�in suggested comp,act·
sions. (If you want 10 specify th� location of
ification as a device for ·hiding· atr� dim�n·
an ant on a clOIMs li=, you would probably
just rq><>rl irs dis�nce z from one end - only
for mum smaller bugs would the ;I�imuth;1i
position � be of inlerest Or importance.)
t The anthropic principle holds that th� laws
and parame� of physics are what they ;Ire
·beau,;.,· (if thaI's th� right word) if they wer�
diff�"'nl, human lif� wQUld be impossibt.:
;lnd we wouldn·t be he", to discover them.
12.5 Dark Ml1lUrjDork Energy
,------,
•
IIU I
II I I I
4
"
.'
1 11
1 11 , . I
6
f
10
s
Radius (kpc)
Fig 11.6 Rotation curv� for th� galaxy NGe 1560. Th� snlid
.
lin� r�pr�s�nts Ih� curv� 10 be ex�cled on the basis of all
observed matter (stars plus gas). [Sourc.: A. H. Broeils As·
lro". l1"d Aslrophys, 256 1 9 (1991),]
the mass ofthe cluster. The result was surprising: 400 times larger than the visible
stars n
i the cluster. Evidently the galaxies contain a lot of matter that does not
radiate (and is called, therefore, dark mailer) [25J. More recently, rOlalien curves
have been measured for a number of galaxies (including our own). These plot the
(tangential) velocity v as a function ofdistance r from the galactic center. Newton's
law of universal gravitation says that for stars well away from the core v should
decrease as l/Jr (Problem 12.10); instead, it typically increases (Figure 12.6). This
suggests that the dark matter permeates a spherical ·halo' extending well outside
the galactic nucleus: Today it is even possible to map out the distribution of dark
matter, using gravitational lensing (the bending of light as it passes through).
So far, though, our only evidence for dark matter comes from its large.scale
gravitational effects, and it is natural to wonder whether perhaps Newton's laws
(and also General Relativity) are n
i correct on some scale, and there is actually no
dark matter out there [261. Short of such a radical alternative, the question remains:
what is this stuff? Could it be ordinary cold matter - sand and gravel, perhaps,
the remnants of extinct stars or dead planets. Almost certainly not. Cosmological
models that are convincingly corroborated by the observed abW1dances of light
elements do not allow for anywhere near enough baryons to account for dark
matter [27J. What about neutrinos? Probably not - even though there are enormous
numbers ofthem, they are much too light to contribute more than a small fraction
of the observed dark matter.t Evidently we are looking for something much more
• The dark malter disCllssed here is nOI 10 be confused with th� missing rrulU' r,",!uireillO 'do",,'
'
the uni�rse. We will talk aboul that in the next section.
t Mor...,�. n�trinos would constitute ·hot' dark matter - they are by nature highly relativistic.
..nd it s
i hard 10 irrulgin� that they could be confi� to gabc:tic halos (or to the primordial ago
gregates from which galaxies emerged).
1 415
"16 1 12 Afterword: Who"s Nut?
massive than neutrinos, but �ike neutrinos} weakly interacting; Bahcall called
them WIMPs (Weakly Interacting Massive Particles)." Their mass is tentatively
estimated to lie in the range 100-100 GeVjcl; they are certainly neutral (otherwise
they would radiate) and stable (left over from the Big Bang). No such particle,
of course, is known to the Standard Model. But supersymmetry does suggest a
candidate: the lightest supersymmetric particle (probably a mixture ofthe photino
and the higgsino - or possibly the Zino - calle<! the 'neutralino' . . . obviously, this
terminology is getting out of hand) is presumably absolutely stable. Large numbers
might be left over from the Big Bang. Another possibility is the axion - the
hypothetical particle introduced to account for the absence of strong CP violation.
But surely the most exciting possibility would be something entirely new and
unanticipated.
How is all this going to be decided? Since the late 1980s a number of WIMP
searches have been under way. They are based on the realization that the solar
system orbits around the galactic center at 220 Ian s-l,t and the earth orbits
the sun at 30km S-I, so we face a 'dark matter headwind' - 135 km S-l in
(northern) summer and 105 km S-l in winter. (Ibe seasonal variation is a lucky
thing, for it should enable experimentalists to filter the signal out of a much
larger - but constant - background due to natura! radioactivity and cosmic rays.)
Several different detet:tion mechanisms have been tried (17), but it is only recently
that their greatly improved sensitivity has approached the requisite level. There
have already been some (questionable) events (28), and convincing evidence may
well come in the next few years. Meanwhile, the LHC should be n
i a position to
create dark matter, and at that point the remaining task will be to demonstrate that
the three approaches (galactic, terrestrial, and accelerator) are all talking about the
same particle [19).
12.5.2
Dark Energy
Before 1998, it was taken for granted that the expansion ofthe universe is slowing
down, due to the gravitational attraction of all matter; the only question was
whether the energy density ofthe universe is great enough to reverse the expansion
completely, leading to a 'big crunch' (see Problem 12.10). Visible matter and dark
matter together amount to about a third of the 'critical density', so for those
who believed the expansion 'should' reverset there was a set:ond 'missing mass'
paradox. unrelated to the dark matter problem: where is all that 'extra' energy?
i ter�ct ,,"ly gravi�tionaUy. but �s ai� (27) remarks coyly. 'If
• In prn
i dpk, d..rk rruttt�r might n
that is really th� use. physicists have no hope of � detecting it' (that is. as individual parti·
d�s). For this reason. at least. it is generally assum� th.:it d..rk matter participates n
i the we�k
t � dark maner halo (since it is only very weakly (oupled to matter) does not (one assumes)
n
i teraction.
share in the g;lt.ctic rotation.
t The widely �(cepted inlbtionary cosmology rtqwirli. th.:it tht total dtnsity of the universe have
exactly the critical value.
126 Conclusion
This problem was turned inside out by the astonishing discovery that the
expansion of the universe is not slowing down at all, but rather accduating.
Evidently Newtonian gravity (universal attraction) is not right on the largest
scale - either that or there is some new force that is repulsive n
i nature and
overwhelms gravity n
i this case. [n General Relativity, there s
i a (sort of) natural
place for an extra term that could account for the phenomenon: the cosmological
constant, A. Einstein's original theory (with no cosmological constant) implied
that the universe expands - something he regarded as absurd. He was able to
rescue the theory by introducing an ad hoc source term, whose strength (A)
could be adjusted to stabilize the universe. (Mathematically, the cosmological
constant introduces a kind of primordial repulsion, or negative pressure, that
balances the universal attraction on a cosmic scale.) Later, when Hubble discov·
ered that the universe is in fact expanding, a chagrined Einstein disowned the
cosmological constant, calling it ·my greatest blunder'. But when the accelerated
expansion was discovered, the obvious remedy was to resurret:t the cosmological
constant [30].
There is, however, a subtle distinction between the original notion of a cosmo·
logical constant and its contemporary reincarnation. Einstein conceived of A as an
unexplained fundamental constant of nature - analogous to Planck's constant or
Boltzmann·s constant; there were tW<l distinct sources of gravitation: matter (actu·
ally, the stress ttnsor, incorporating energy, momentum, and stress of aU forms),
and A. [n the modern version A is taken to have a dynamical origin, n
i the form of
dark tltugy associated with the vacuum expet:tation value of some quantum field.
It is, n
i effect, a constant term in the stress tensor, pervading all space uniformly,'
that we choose to peel off and treat separately. But what the nature ofthis field (or
fields) might be is at this stage a mystery. Worse than a mystery, because attempts
to construct model theories tend to yield values ofA that are 120 ordus ofmagnitude
too great![31] Obviously, we have a lot to learn.
12.6
Conclusion
Most particle physicists anticipate that the LHC will produce Higgs hosons. Many
believe it will create the first supersymmetric particles. Some think it will yield
evidence ofextra dimensions. Perhaps. But there is another possibility that very few
take seriously: substructure - the idea that quarks and leptons (and maybe also the
mediators) are composite particles, made of even more elementary constituents.
This would change everything, just as the quark model changed everything 4()
years ago, and Rutherford's atomic model changed everything a century ago. In
any event, we almost certainly stand at the threshold of a fundamental revolution
in elementary particle physics [32J .
• This is in contrast to dark mall..r, which is concmuated in galactic halos.
1 '17
418
1 12 JJJklWOrd: Whot's NexI?
References
1 See, for example, the special
section in (2007) Science, 315, 55.
2 The Higgs mechanism was prOtrned
independently by Englert, F. and
8rcut, R. (196-4) Physital Review UI·
lers. 13. 321: and (a) Higgs, p. W.
(196-4) PhY'ics ullt.., 12, 132; (1964)
Physiall Review Wltrs, 13, 508.
3 The 'bible' for this m<lterial is
Gunion, J. F. et al. (1990) J1w: Higgs
Hunl4:r's Guide, Addison·Wesley, Red·
wood City. CA.
( LEP had already exduded a Higgs
boson with mass below 114 GeV/cl:
the statistically marginlll obser·
vation in 2000 suggests a mass
of 115 Gevlc1. See, for example,
Renton, P. (2004) Nature, 428, 141.
� Estim.lles of the Higgs mass are sen·
sitive to the exact value of the top
quark mass, and of course they de·
pend on the model - supersymmetry
puts an upper limit on the
mass of the lightest Higgs par·
tide at 140 GeV/cl. See. for in·
stance, Schwan schild, B. (Au'
gust 2(04), Physics Today, 26.
6 For an accessible introduction see
Peskin, M. (2(06) Physics (SO lec·
ture not�, Fall 2006, Stanford
University, available on·line.
7 Georgi, H. and Glashow, 5, L. (1974)
Phy.ical Review Lttte", 12. 438.
s Sruozawa, M. et al. (1998) Phys'
icaI Rtlliew UUers. 81. 3319.
9 For a comprehensive review see
Ross, G. (1985) Grand Unified J1w:.
ories. Perseus, Reading. MA; (a)
Langacker, P. (1981) Physics Reports,
n, 18S.
HI 't Hooft, G. (1974) Nuclear Physics,
..kov. A. M.
879. 276: (a) Poly
(1974) JETP Wl4:rs, 20, 194.
11 Cabrera, B. (1982) Physiall Review
Latus, 48. 1378.
12 For an account of the search
for stray antihelium from some
distant antigalaxy see Barry, P.
(2007) SciellCe News. 171. 296.
U Sakharov. A. D. (1%7) JETP !Ltft..,
:
5, H.
I( See, for example, the article by
Kirkby, D. and Nir, Y. (2006) Re·
view of Parlic/t Physics. 146.
IS See. for instance. Peskin, M. (2002)
Nature, (19. 24; or (a) Harrison.
P. (/u1y 2001) Physics World, 27.
16 Peccei, R. O. and Quinn, H. R.
(1977} Physics Review LtII4:Il. 38,
1440; and (1977) Physiwl Review
D, 16, 1791. For a delightful ac·
COWl! of the Quinn/Pccrei mecha.
nism see (a) Sikivie, P. {December
1996) Physics Today, 22; for the cur·
rent state of;uion searches, see (b)
van Bibber, K. and Rosenberg. L. J.
(August 20(6) Physi.:! Today, 30.
17 Wess, J. and Zumino. B. {197()
Physics !Ltftll B, 49, 52.
18 For an excellent semi·popular ac·
count of supersymmetry. see Kane.
G. (2000) SUpersj'l!lmet')': Un"eu.
ing the Ultimate Laws of Nature,
Perseus. Cambridge. MA; for a rca·
sonably accessible introduction to
tm, temnical details see A SUJ'<'r·
symmetry Primtr, by S. P. Martin,
on the web al hep-phf9709356.
19 See the article by Haber. H. E. (2006)
RPP, p. 1105.
:to Brown. H. N. ct al. {2(01) Phys·
icaI R&I'lew Wltrs. 86. 2227.
21 Schwarzschild, B. (February 2(02)
Physics Toda�, 18.
U For a thrilling sem
i·popular his·
tory of string theory, see Greene,
B. R. (1999) J1w: Elegant Universe.
W. W. Norton, New York; for an
accessible introduction to the the­
ory itself. see (a) Zwiebach. B.
(200() A First Course in SIring J1w:.
ory, Cambridge University Press,
Cambridge. UK; the standard grad·
uate textbook is (b) Pokhinski. J.
(1998) String Theo,)" Cambridge
University Press. Cambridge. UK.
21 For an accessible account nf �tra
dimensions see Randall, L. (July
2007) Physics Today, 80. For the sta·
tus ofsearches for �tra dimen·
sions see the IIrticle by (a) Giudice.
G. F. and Wells, ,. O. (2006) Re·
� of Parlick Ph�sic.!. 1165.
12,6 Concll1sion
24 In 2006, there Were signs of a
matter, see Cline, D. B. (March
backlash against string theory
2S Schwuzschild, B. (August 2007}
2003) Scientific Ameriron, 51.
and its p"'rvasi� influence on
:
particle physics Richter, B. (Oc'
tober 21)(16) Physics Tod!1Y, 8; (a)
Smolin, L. (21)(16) The Trouble With
Ph)'Si<:s, Houghton Mifflin, New
York: (b) Woit, P. (2006) Not Even
Wrong, Basic Books, New York.
Physic. Today, 16.
2� Peskin, M. E. (2007), journal of tlu:
Physical Society ofj!1p<Jn, 76, 111017.
lit This is not the only possibility;
for an outstanding review see
the article by Turner, M. S. and
Huterer, D. (2007), journal oftlu:
PhY'icaI S"'�ty ofjap<Jn, 76, 111015.
11 For an accessible aplanation
see Adler, R. J., Casey, B. and
l5 Zwicky, F. (1933) Hdvetict> Physics
Act!1, 6, 124. For a masterful re­
view of the first 55 yeus, see (a)
Trimble, v. {1987} Annual Review of
2' MilglOm, M. (August 2002) Scientific
Astronomy and Astrophysics, 25, 425.
Jacob, O. C. (1995) Ameriwn
journal of Physics, 63. 620,
12 Ellis, ,. (21)(17) N!1/Urt, 448, 297.
II Planck, M. (1899) Sitzungsber. DIsch.
Akoul. Wis$. Berlin-Moth·PhY'. Tech.
Americ<ln, 42; (March 21)(17) SC�IIU
News, 206.
27 For a beautifully deu SUIVey of
dark muter candidates, and a·
Kl" 440.
perimental searches for dark
Problems
q .. g....;1fi1iii. Thus, confinn Equation (12.1)
IblUse Problem 10.21 and Equation 10.130 to determine the �rta factor for the
12.1 (al Use Equation 10.132 to determine the mass of the W, in terms of v _ IA/). and
coupling ofthe Higgs to a quark or lepton.
1(1 Use Equation 10.136 to detennine the verta factors for the couplings "WW, "22,
12.2 (a) Calculate the decay rate for " ...... f +J (wheref is a quark or kpton), in the MSM,
,rul "'"
(b] lf mh - 120 GeVIr?, what are the branching ratios r(bb)/r(" ) and r(bb)/r(r+ r-)�
(Include a factor of3, for c% r, in the case ofquarks.]
12,1 (>1 Calculate the decay rates for " ..... w+ + W- and " ...... 2 + 2, in the MSM.
,
(�)
1 _ 4 Mw + I2 Mw 1 _ 4 Mw
w
�
�
m�
M
'( , ' ) [ ' ]'"
, ( )'( , '" )[ ']"']
r( w+w- ) = a",mhc
1M
a",mhc
mh
r(ZZ) = --
32h
Mw
m
m
MZ
1 - 4- + 12Mz
m�
t
m
(hi If m� _ 120 GeV1,1, wh<lt is the ratio r(w+ W-)tr(2Z)?
Mz
1 - 4m�
1 419
420
1 12 1{1�NX)rd:
Whal's Next?
11.4 ESlimatt the long�t proton lifttimt that could � me;o.sured in a realistic laboratory
experiment. [Hi"': How many prolons could you sample, in a practial experiment
(Super.K, for instance)? How long would you (or - mOre to tht point - your funding
agtncy) k prepared to wait?)
U.S Estimate the lifetime of the proton, n
i the Glashow/Georgi model.
[Hillt: Don'llry to cakukltt anything here - you don't have anywhere near enough infor·
mation. The real question is how 1M lifttime formula depends on the various masses.
Study other decays - the muon, the neutron, the pion - and exploit dimtnsional
analysis. if it helps.)
12.6 Derive Equation (12.9). from Equations (12.?) and (12.8)
12.7 Consider vectors in the xy plane:
Ii) Show that a (counterclockwise) rOlalion [) carri� a vector a • I"., (ly) into a' • (a'x.
'
,, y) given by
a� = cos9a,, - sin9a.,.> a� = sin9a. + cos9ily
(b) Show that the dot product of two VK!ors is invariant under such a rotation: a' . b'
. a · b.
1<)Now consider an ;"Ji";Itsimal rOlation de. Expand the transformation rult in (a) to
first order in dO.
(d)Show that the dot product is invariant (to first order) under infinitesimal rOlations.
[Of course. if you already know it's invariant underJi"iu tr.msformations. the proof
for infinilesimal transformations is redundant The point is that the infinitesimal
ase is typically much simpler,[
12.8 The purpose of this problem is to prove that the action described by the Lagrangian in
Eq. (12.11) is invariant under the supersymmttry transformations in Eq. (12.10).
(aJ Show that J,p0 = itf and 61i (i/�)fy�la�41°).
=
(blConsider firsl the scalar 'kinetic' term, L, = t(a"4>')(a,,4» ; show that U, =
«l"4>)(a,,1i)€ +�«l"4>°)(a,,1/t).
(e) Next treat the spinor 'kinetic' tenn. .cl im�r"{a ....). Show that J.cl -J.cl +
"
a" Q", where Q" 1i{a"4» l + t...,...·[4>·{a.t) - (a.4>·)"'). where .,.... is defined in
=
=
III
Eq. 7.69.
(d)Now examine the mass terms, .c) = - 2 (=/1I)1,p°41 and .c. = _1IIC1 �",. Show that
JL) = -(=/II)l(1il4> + ,po..",) and U., = i(IIIC/II)[-iy"(a..41°)t + 1iy"((a�,p)).
(e)FinaUy. invoke the Dint equation. which follows from the Euler·lagrange equations
�Y"l4l1·
{Eq. IO.IS), 10 show thai J.c.
=
-J.cj + a" R", where R" = i(=/II)[-iy"4>'''' +
i
Although the full Lagrangian (it' = Zi + it'l + 2) +�) s nOI invariant. it changes
only by a tOlal divergence. Jit' = a" IQ" + R"), so the action and the equations of
motion art n
i variant Notice, however, that tht scalar and the spinor have to arry the
"'''''' mass forthis to work.
12.9 (>] From c, II, and G (NeWlon's constant of universal gravitation). construct a quantity
i
If with the dimensions of length. a quantity If with the dimensions of tme.
and a
quantity mp with the dimensions ofmass. These are known as tht Planck ltngth, the
Pklm;k lime. and the Planck mass. respectively. afler Mall Planck. who first published
"umbe�, in meters. seconds, and kilograms. Also calculate the Planck e",,'l!Y (ep .
m,.cl), in GtV. [These quantities sel the scale at which quantum gravity is expecled
them in 1899 - the ytar hifortthe eponymous constanl itself[33]. Work out theaclual
to be relevant]
[b)What is the gravitational analog to 1M fine structure constant? Find
the actual
number, using (i) the mass oflM electron, (ii) the Planck mass.
12.10 Find the velocity v as a funct on of orbital radius r. for an object in a circular trafectory
i
around a fixed center ofmass M (for example, a plantt about the sun).
12.6 Cone/usion 1421
12-11 A quick naive way 10 calculate the critical density is to picture the universe as a unifonn
sphere of radius R, and set the e$Ca� vdodty for a p.article at the surface equal to the
expansion velocity {from Hubble's law). v . HR. On this basis. show tlul
Pc = 8JrG
lH'
Look up the value of Hubble's constmt {H). ;md determine the critical density. in
kgfm-1.
Appendix A
The Dirac Delta Function
Introduction 10 t� Dirac deltafunction
i an infinitely high, infinitesimally narrow spike at
The Dirac delta function, S(x), s
0)
the origin, with area 1 (Figure A.I). Specifically
SIx) =
f 000,
1 ,
if x #
ifx = O
,nd
(A.I)
Technically, it's not a function at all, since its value is not finite at x = O. In the
mathemalical literature it is known as a genualiztd function, or distribution. It is,
if you like, the limit of a stquence of functions, such as rectangles of height n and
width lIn, or isosceles triangles of height n and base 2/n (Figure A.2), or any other
shape you might wish to use.
Iff{x) is some 'ordinary' funcion
t
(that is, not another delta function - in fact,
just to be on the safe side let's say thatf(x) is continuous) then the productf(x)S(x)
s
i zero everywhere except at x = o. It follows that
f{x)6 (x) = f{O)S(x)
(A.2)
(This is the most important fact about the delta function, so make sure you
at x = 0, we may as well replace fIx) by the value it assumes at the origin.) In
understand why it is true. The point is that since the product is zero anyway except
particular
(A.3)
Under an integral, the delta function 'picks out' the value of f(x) at x = O. (Here
and below, the integral need not run from -00 to
+00;
it is sufficient that the
domain extends across the delta function, and -� to +" would do just as well.)
l,wed""lioM 10 Ele""'1IID1)' hnides, Second Edilion. !}avid Griffiths
Copyrighte 2008 WllEY·VCH VerLog GmbH & Co. KG.A, Weinhtim
ISBN: 978·J.S27·4-0601·2
<124
I
A The Dirac Della Function
6(lC)
,
Fig. 1<.1 The Dirac delta function (you must imagine. how·
ow)
e><er, that the spi�e is 'nfinitely high and infinitesimally nar·
r
.
Ofcourse, we can move the spike from x = 0 to some other point, x = a:
(A.4)
6(x - a) = (0, iffi xx "#= ala ,nd i:6(x - a)dx=1
(see Figure AJ). Equation A.2 generalizes to
(A.S)
f(x),)(x - a) =j(a),)(x -a)
and Equation A.3 becomes
(A.6)
i: - a) dx f(a)
00,
=
j(X)8(X
How should we interpret the expression fi k is some nonzero (real
number? Suppose we multiply by an 'ordinary' functionj(x) and integrate: )
i:f(x}6(I:x} dx
We may change variables, letting y so that x rlk, and 11k dy. If k is
positive, the integration still runs from to +00, but if k si IUgative, then x 00
8(h),
.. h,
=
-00
,
1 14 I n
,
112
J X
Fig. 1<.2 Two sequences or functions whose limit is �(%).
dx =
=
A The Dim, DelIo Fu"ctio"
t 6(x-a)
I
I
I
Fli. A..l 'Gr�ph' of �(x - 41).
.
,
impli�s y = - 00, and vict vtrsa, so tht limits ar� rtverstd - restoring th� "proper"
order costs a minus sign. Thus
IA.�
(Th� lower signs apply when Ie is negative, and we account for this neatly by putting
absolut� value bars around th� Ie, as indicattd.) In this context, then, S(Iex) serves
th� same purpose as llllleI)S(x):
d,
(00 fl'I" "'I"" ,-co
(00 ft'l [+"'1]
,-co
I I
�
(A.S)
B«aus� this holds for any fIx), il follows that tht dtlta function exprtssions are
equal;'
I
Sllex) = ]Ie] SIx)
(A.9)
What we have just analyztd is really a special case of the general form S lglx)),
where g(%) is som� function of %. In g�n�ral, SIg(x)) has spik�s at th� ztros, %L, %2,
Xl, . . . , of g(x):
g(x;) = 0 (i= 1,2,3, . . . ,n)
, You oughllO ponder th�1 LIII *tq> for � mo­
�nt. Ordirlilril)l. the equality of two integrals
certlIinl)l dMs ,,01 imply equality of th� in�
grand.•. The crucial point hen! i. th�1 the in�
grail are KjUllI for allY/(1<). Suppose the ddta
funCtion expressions &(h) and (Itlkl)8(1<) ",.
IWIlly t1ifftrnJ. SlIy. in the nfighbofhood. of the
point % _ 17. Then t wouJd pick, function
/(>4 �t wu sharply pt�ktd �bout " '" 17.
(A.IO)
ind tilt intta"'ls would not be KjwoL Sinct,
on tilt (onlTaIY, Ihe integrals mUll be eqwol.
il I'oJIows that the <kIt. function expressions
are therT\#J_ eqUllL Well, lIeC.hnkally thry
might still differ at isolat..d points, provided
the� contribut� nothing 10 the in!tgral, but
we can *ilm� this objection by noting tmt
both ,ides of Equation A.9 �rt oorly ttrO a·
Ct]M lt J< _ O.
I <425
4261 A Th�
Dim, �!ro Fundion
In the neighborhood of the ith zero, we may expand g(x) as a Taylor series:
g(x) = g(x;} + (x
�
Xi)g'(X,) + } (x - Xi)2g" (Xi) + . - :;;: (x - Xi)g'(X;)
(A.ll)
In view of Equation A.9, the spike at Xi has the fOrm
J(g(x)) =
1
'
J(X- Xi)
)g (Xi))
(x:': Xi)
(A.I2)
The factor 18'(XI))-1 tells us the 'strength' of the delta function at Xi. Putting this
together with the spikes at the other zeros, we conclude that
(A.B)
- Xi)
L --.5(x
!g'(Xi) )
•
.5(g(x)) =
1
i.1
Thus, any expression of the form .5 (g(x)) can be reduced to a sum of simple delta
functions:
Simplify the expression .5(x2 + x - 2).
EJ(ampl� A. J
Solution: Here g(x) = x2 + X - 2 = (x - I)(x + 2); there are two zeros, at Xl = 1
3 and g'(X2)
-3.
and X2 = -2. Differentiating, g'(x) = 2x + I, so g'(X1l
Thu,
=
1
1
J(x2 + x - 2) = 3 .5(x - l) + 3 8(x+2)
A.4):t
10,
=
M
Il
The Dirac delta function can be thought of as the derivative of the Heaviside step
function (Figure
O(x) ;;;;
I,
(x < o)
(x> 0)
(A.14)
Obviously, dOIdx is zero everywhere except at the origin, while
- dx = 8(00) - 8(-00) = I - ° = 1
dx
100 dB
_00
(A.IS)
so d91dx satisfies the defining conditions (Equation A.I) for 8(x).
It s
i an easymatter to generalize the delta function tothree (ormore) dimensions:
(A.16)
• Equltion A.ll s
i t:wct. notwith.sJanding tilt truncatrd Taylor =ies (Equation A.ll) I w;rd in its
d�rivation. At Xi, tht '�xtra' \l!rntS ar� �ero, sinc� Ihey conJain powers of (x - Xi).
t Th� valu� at th� discontinuity �ldom matt�rs. but if it worries you. d�fin� 0(0) _ 1/2.
A Th, Dirac Deihl FunctiQn
/I(x)
x
Fig. A.<4 The Heaviside thet� ('step') function.
This three-dimensional delta function is zero everywhere except at the origin,
where it blows up. lbe triple integral over �l (r) is 1:
,nd
(A.IS)
For example the charge density (charge per unit volume) ofa point charge q located
at the point ro can be written as
,
p(r)
= o¥l (T - TO)
(A.19)
Problems
A.I laf li(2,.2 + 7x + l)�(x - If dx
1
=
(hJ Ii In(l + x)�(". -x) dx = 1
A.2 Use Equation A.ll to simplify the expression �(..rxr+1 - " - 1).
A.3 Use Equation A.13 to simplify the expression 8(sin ,,). Sketch this function.
A.4 Letf(f) = I; �(y - ,,(2 - xJ) dx. Findf(y). and plot il from y = -2 to y = +2.
[;& - 3)1 dx 1 [Hint Integrate by parts.)
gni (10 5 significant digits)
A.S I�l x'
�(x
A.6 EvaJuate the inte
=
{I 9(2x <4)e x dx
i_.
_
-l
A.7 Evaluate I r · (a - r)�)(r - b) dt" if a (1, 2, 3), b
over a sphere ofrlIdius 1.5 centered at (2, 2, 2).
=
=
(3, 2, I), and the integration is
1421
"
I'
Appendix B
Decay Rates and Cross Sections
Summary offormwasfor decay rates and scattering cross sations.
B.l
Decays
Suppose particle 1 det:ays into particles 2, 3, 4, . . . , n:
1 ...... 2 + 3 + 4 + "
'+n
The decay rate is given by the formula
dr =
141\n:l {[ {;n�:��2 ] [(;H�:��) l·· [(2:�:�EJ}
X(br)�,,4(Pl - P2 - Pl - . . . - p�)
(B.l)
where Pi = (E;!', Pi) is the 4·momentum oflhe ilh particle (which carries mass mj,
CJPT
=
so Ej =
+ mfc2). The det:aying particle is presumed to be at rest: PI (mle; 0);
S is a product of statistical factors; Ifj! for each group of j identical particles in the
final state,
B.l.1
Two-body Oecays
If there are just two particles in the final state, the integrals can be performed
explicitly. The total deuy rate is
(B.2)
where Ipl is the magnitude of either outgoing momentum:
'
- 1m· + m! + m4
Ipl = V ] -.
1
2m]
_
2m2"J
] -, 2m2m2
] J 2"Jm2
--, J
_
_
lnlrodu«i"" 10 Eltm<nlO'l' Pdnidu. Socond Edilion. David Griffiths
Copyright C> 2008 WlLEY·VCH VerlagGmbH & Co. KGaA. Weinheim
ISBN,978-J.S27·-406Q1·2
(B.3)
430
I8
Dealy Rotts ond Cross Sectjonf
In particular, ifthe outgoing particles are massl(.SS, then I pl = ml e/2, and
r�
s
161Tfiml I..KI
--
,
(B.4)
'.2
Cross Sections
Suppose particles 1 and 2 collide, producing particles 3, 4, .
. .
, n:
1 + 2 ...... 3 + 4 + · · · + n
The cross section is given by the formula
(8.S)
where (as before) Pi = (E;!c, Pi) is the 4-momentum of particle i (mass mil,
l
E, = e mfc2 + Pt, and S is a statistical factor (Ijj! for each group of j identical
particles in the final state).
B.2.1
TWO-body Scattering
If there are just lwo particles in the final state, the integrals can be performed
explicitly.
(a) In llu Clnter-ofmomenlumfratm
>fld
IB.�
where I p,l is the magnitude ofeither incoming momentum,
and IPfl is the magnitude of either outgoing momentum. In
particular. for elastic scattering (A + B
so. letting E ;5 (EI + E2)f2:
cia
dO
=
(",-)2 51.412
16n
- A+
B.2 Cross $u1ions
BJ. Ip;l = IPrl.
(8.8)
E2
(h) 111 1m labframe (particle 2 at rest)
(B.9)
In the case of elastic scattering (A + B _ A + B).
IPl l EJ cos 9 1
(8.10)
If. in particular. the incident particle is massless (m}
reduces to
= 0). this
(B.ll)
Ifthe target recoil is negligible (m2c2 » EI). then
Equation B.IO reduces to
(B.12)
If the outgoing particles are massless (m3 = m4 = 0).
Equation B.5 yields
1-431
Appendix C
Pauli and Dirac Matrices
Pauli and Dirac matrices.
C.l
Pauli Matrices
The Pauli matrices are three Hermitian, unitary, traceless 2 x 2 matrices:
(e.l)
four-vector, andwe do not distinguish upper and lower indices: CTI = CT I , CTl, = CT2,
(Often we use numern:al n
i dices: Ul = CT",CT2 = CT�,U) = Uz: CT is not part of a
CTl = CTJ.)
(a) Product ruks.
(c'2)
(A 2 x 2 unit matrix is implied in the first term, and
summation over I: in the second). Thus, in particular:
(0)
[CTi,CTj) = 2��1;<7l
(CTi,CTj) = 2��
(e.4)
(commutator)
(e.5)
(anticommutator)
(e.6)
and for any two vectors a and h,
(a CT)(b'CT) = a . b + io" . {a x b)
I",rod"",;"" ,., £/tmmJary Ami&.. Sro:>tId flli.;"", David Griffiths
Copyright 0 2008 WILEy.vCH Verlog GmbH & Co. KGlA-, Weinheim
ISBN: 97S-H17-40601·2
(e.7)
C Pawli andDirac Mamers
(b) Exponlntials.
eiiJ." = ws O + ib · u sinO
(CS)
Dirac Matrices
C.2
( )
(0 )
0
The Dirac matrices are four unitary traceless 4 x 4 matrices:
'� 1
y 0
0
-1
;
.
y'
-u;
'=
"'
(C9)
0
is the 2 x 2 matrix of zeros ; .,. i are the Pauli
matrices. Lowering indices changes the sign ofthe 'spatiai' components: Yo = y l1,
Yi = -y'.) We introduce as well the auxiliary matrices
(Here 1 is the 2 x 2 unit matrix, and
(CIO)
(CII)
(Cll)
For any four-vector al' . we define the 4 x 4 matrix I- as follows:
(CU)
(a) Product rules. In terms of the metric
(1 0 0 1.)
-1
0
0
0
-1
0
r-
(C14)
(note thatgl'"gl'" = 4), we have:
y l' y
' + y"yl' = 2g"',
(CIS)
Yl'Yl' = 4
YI' Y·Y" = -2y",
y..y"y�y" = 4g"�,
(CI6)
y..I-r" = -21-
y../�Y" = 4a· b
y"y. y� y " y" = _2yay�y.,
(CI?)
(CIS)
Y"I-�'Y" = -2,$1-
(C19)
C.l Dirac Mauicts
(b) Trace theorems. The trace of the product ofan odd number of
gamma matrices is zero.
Tr(l) = 4
(C.ZO)
Tr{y"y") "" 4g"".
Tr(JijS) "" 4a· b
Tr(yl' y"yl,y�) = 4{i'"gM g"l.g"tJ + y}'tJ g'l,).
(C.ZI)
_
TrlP$,j) = 4 [{a. b){c . dJ - (a . c)(b . dJ
+(a· dJ(b· e)l
(C.ZZ)
Since y5 is the product of an even number of y matrices. it
follows that Tr{ySy") O and Tr{ySy"y"yl.) O. When yS is
multiplied by an evm number of y 's, we find
_
_
Tr(ys) = °
(C.23)
Tr{ySy"y') = 0,
(C.Z4)
Tr(y5y"y"yl.y") = 4i€,,'1.<r.
Tr{y SIJ51j) = 4i€,,"1.<ra" b,cl.do-
(C.Z5)
where E'",l,,, = -1. if J.tLl.l.O is an even permutaion
t
of 01Z3.
+1 for an odd permutation, and ° if any two indices are the
same. Note that
(e) Anticommutation rel6tions.
(y.... y" ) = Z8"".
(C.Z6)
(C.Z7)
1 435
Appendix 0
Feynman Rules (Tree Level)
Ftynman rulesfor QED, QeD, and IV((.II: interactioltS.
0.1
External lines
Spin 0 : (nothing)
I1 . j
�
Spin ;
Spin
.
Incoming particle : u
Incoming antiparticl e : ii
Outgoing particle : u
Outgoing antiparticle : u
incoming : E"
outgoing : E;
0.2
Propagators
�
=
,,
�
Spin O : "
'I" (/>te)2
Spin ! :
i(d+ me)
q2
(me)2
Massless : -ig."
T
Spin 1 :
Massive :
-i!g.,. - q"q./{me)2 )
q2
{me)2
/n!rod",!i<Jn I<> Ek""nla'l'Partida. Smmd £di!i<Jn. D�vid Griffith$
Copyright 0 2008 W1LEY-VCH V�rJ.ag GmbH &Co. KG.A. W�inh�im
ISBN: 97&.3-S27--10601-2
.3a
I D FeyrlmQrI
R"'�I (Tne Level)
D.3
VerteJl Factors
QED:
ig..y "
(g. = .J4:rrrt)
QeD:
.,
.
-i
a.
-2""' >.."
y"
_gJa/iY [&... (ql - q* + g.,d� - q))"
+g..,,(q) - qJ ).]
-
_ig;lfatlqfy6� (g,,�g.p g"pg.l)
+rOQjtlrq(&,.8l.P g,,�gyp)
+ryqf·tlq(g,.pg.� - g".84» )
_
GWS:
w-
�Y"(l - yS) (Here I is any lepton, and
v!
r
,
the corresponding neutrino.)
D.J Verlex Faclof'5
�Y"(l -
yS)Vq (Here i or I,and) d,
or b; Vis the CKM matrix.)
= u, C,
-
=fy"(.{. �yS)
<.
<,
v" LI", V,
l
,
,
U, C, t
d, s, b
"
"'I
Z
Z
W'
Z
"
,
X"
w-
w�
"
.<
-i + 2sin22 e.. -�
� - � sin /;l... l
-t+i sin2e", -
,
,
ig..cos9",[g"J.(ql -q2)"
+g,.,,{q2 -ql). + &..(ql -qd.d
X"
W-
Here! is any quark or lepton;
Cv and CA are given n
i the
following table:
f
r-.J.t
w-
W·
,
=
S,
1 439
«0
I D Feynma� Rultl (Tree Level)
The weak coupling constants are related to the electromagnetic coupling constant;
g... -
�
"
g. - sin liw cos e", .
.
sin 8.., ,
There are also 'mixed' couplings of the photon to the W and Z:
+g.,,('12 - '13)" + g"o(q) - qllJ.]
ig.[g"J. ('11 - '12)"
X'
WW·
"
Y
Y
"
Index
•
A: sn Baryon number
ABC th�ry 211-223
Abelian 118, }65
- Set also G�uge inV<lmnce; Groups
Acccler�tor 4-7,401
Additive quantum nwnber 141
Adjoin! 236,361
Alloww 120, 127, 161-162
0; see Fine structure constmt
0,
173
308, 315
cr particle 5, 14, 156, 388-389
Amplitude 132-133, 156, 203-204,
211-214, 244
Anderson, C. D. 19, 21. 31
Angular momentum 120-128
- addition 122-125, 154
cr.,
- eigenfunctions: see Spherical
harmonics
- eigenvalues 120-121
- matrices: see Spin matrices
- orbital 120-121. 161-163, 182
- spin 120-121
Annihilation: It< Pair annihilation
Anomalous magnetic moment
- electron 18, 165, 167,246,266
- muon 413
- proton 168
Anthropic principle 414
Antibaryon 37.40
AnticommuUtor 155,228,237,252,435
Antielectron: see Positron
Antimatter 21,23
Antimeson 37.41
Antineutrino 24.27.128
Antineutron 21-22
An!ip�rtide 3. 20-23.37. 39.61-62.
230-234
Antiproton 21, 33, 106-107
Antiqu�rk 39,49
Antiscreening 300-301
Antisymmetric sute 122. 125. 183-184
Antisymmetric tensor 110-111. 253
Antisymmetri:ution 245
Associated laguerre polynomial 162
Associ.,ued production 34
Astrophysics 7, 394, 396. 400-401
Asymptotic freedom 68-70,83, 220,
298-301
Atmospheric neutrino 394-395, 399-400
Atomic number 24
Atomic weight 24, 3.87
Axia! vector 139-141,3OS
Axion 411,416
A:timutha! angle 201
b
B; see Bottom
B factory 47,148
B meson 47.148-149
b quark 47
Bahcall./. N. 389-390, 416
Bare: Stt Charge; Coupling consunt; Mass
Barn 211
Baryogenesis 411
Baryon 19,30,32, 35,40,44,122-123,
180-193
- decuplet 36.40, 122-123
- magnetic moment 189-190
- mass 191-193
- number 22, 33, 81, no, 408-409
- octet 35.41, 122-123, 188
Beautiful baryon 47
Beautiful meson 47
Beauty 47,49
- Set (lisa Bottom
p: stt Electron; Positron
inlroduai"" 10 Eltmtnf<ll)' Panicle<, SwwJ Ediricfo. D...id Griffiths
CopyrlihlC> 2008 WlLEy.vCH Veri,s GmbH & Co. KG.A. Weinheim
ISBN: 978·3·527-40601·2
4421 'l1du
Beta decay 23-29, 47-48, 56,76,
136-137,318
Bethe, H. A. 166, 219, 388
Bev;o.tron 21, 106
Bhabha scattering 62. 86, 246
Big B�ng 71.409.416
Bilinearco\';\ri�nt 235-238
Bindingenergy 103, 159-160. 194
Bispinor 229
Bjorken, j. D. 44. 228, 233
Blackbody 15
Bohr. N. 15.24.162
Bohr energies 162
Bohr m�gneton 190
Bohr model 15, 163
Bohr radius 162
Boson 122,141, 183
Bottom 47
Bonomonium 159. 175-177
- See also '( (upsilon) meson
Bound state 2, 159-193
Branching ratio 79,198. 404-405
Brane 413
Breit frame 112
Bremsstrahlung 403-404
Brick WOlll frame: 5U Breit fr.tme
Broken symmetry 135-136. 375. 378.
402,408
Brookhaven 29, 33,44,71, 147
Bubble chamber 7
,
C: see cJu.rm; Charge conjugation
44-45
Cv. ',t 320,331-332
Cabibbo. N., 77 324-327
Cabibbo angle 321, 324-325, 329
Cabibbo theory 324-328
Camouflage 300
Casimir's trick 249-254, 270
Cathode ray 13
Center ofmomentum 64, 106-108
Central potential 161
Centrifugal barrier 161
Cerenkov radiation 7, 392
CERN 6-7,42. 48,51, 72, 331-332,403
Chadwick- I· 15,24
Charge 148,165, )01
- b;o.re 69-70
- conservation: 5U Conservation l;aws
- effective 68-70,266, 29'.l
- eltttrc
i
70.81, 408--409
- exchange 132
- independence 130
c quark
- quark 39
- renormali;:ation 219-221,265-267
- _ak 71-72, 320
- See also Coupling constant
Charge conjugation 142-149
- conservation 143
- number 143
- operator 142-143,269
- photon 143
- positronium, 171
- violation, 143
Charged weak interaction 72-78, 307-308,
324-329
Ch;orm 44-45,49
Charmed baryon 45-47
Charmed meson 45-47
Channonium 159, 172-177
- Su alw '" meson
Chir.t.l stilte 340
Chromodynamics 59-60, 66-71, 366-369
- Su also Quantum chromodynamics
Circular polarization 241,260
CKM matrix 51, 77-78, 81, 148, 328-329,
350,397,409-410
Clebsch-Gonhn coefficients 124-125, 154
Cloud chamber 7.30-31
CM 64, 106-108
CNO cycle 388
Colermm-Gbshow formula 57
Colliding beams 5, 108-109
Collision 87, 100-102
Color 43-44,49-50,66-67,70, 187-188,
278-279
- current 369
- factor 173, 289-294, 304
- field 369
- octet 187.285
- sextet 293
- singlet 187, 285
- state 284-286
- SU(3 ) 187, 285, 366-369
Colorless p;!rtide 43,285
- See also Color singlet
Commutator 155
Comp;!ctification 414
Completeness 230,2)4, 242-243, 270,
308, 348
Composition rule 40
Compton, A. H. 17
Compton
- scattering 17-18, 2),62, 86, 1l3,
246, 249, 271-272
- wavelength 17,69
Confinement 42-44, 71-72,289
I"dn:
Conjugating pha� 410
Conservlltion laws 79-84
- angular momentum
119, 122
- � system 145-149
- violation 51,139, 147-149,328,
26, SO, 117,
- b;"ryon number 33,79,81,85
- charge 79,81, 85,117,238, 270
- color 67,81,85
- el�tron number 29,75,81
-energy 24,64,SO,87, 100-IOI, 117,
213, 244
- flavor 75,82, 324
- isosopin 130
- lepton number 28-29, 7'.1,81,85
- mass 2, 100-102
- momentum 64, SO, 97-101. 117,
213, 244
- p.arity 73, 141-142,309
- muon number
29,75,81
- quark number 81
- stnmgeness 34,82, 324
- tau number 75,81
Conserved current 238, 270, 358
Constituent qu;ork 58, 180, 190, 193
Continuity equation 238, 270, 358
Continuous symmetry 375-377
Contraction ofindices 96
Contraction theorems 252-253
Contravariant 94
Cosmic ray 4-5,19, 30-31,110, 394, «11
Cosmological constant 417
Coulomb force 1,9, 18,61-62,73
Coulomb gauge 2«1-243, 286, 307
Coulomb potential 162-163,202,290,334
Coupling constln! 67,84-85, 213, 3(6
- bare 220, 265
- dimensions of 213
- df�tive 220, 265
- el�tromagnetic 84-85, 244
- physical 219-220
- renonnaliud 219-221. 265-266,
298-301
- running
68,85, 220, 265-266,
300-301, 405-406, 412
- strong 67,84-85,283-284
- weak 84-85, 308, )l5, 320, 332, 346
- Stt also Charge, Fine structure
constant
Covariant 94, %, 226, 268
Cnvariantderivative 360, 363, 367, 378
COWlln, C. L 27
CP 144-149,«19-411
- IfJ system
- eigenstate
- invariance
148-149, 409-411
146
145-147
409-411
Cronin,). W. 146-149,409
Cross product 368.408
Cross section 132-133, 156, 199-203, 209,
430-431
- A + A ...... B + B 215-216
- hard sphere 200-202
- Mott 245-246, 254-255
- nucleon-nucleon 132
- pair annihilation (QCD) 198
- pairannihi
btion (QED) 261
- Rutherford 202. 2S5
- Suolso Golden rule. Scattering
Crossing symmetry 22-23,62
Current
- charged weak 341-346
- color 369
- con�rved 358
- electromagnetic 238.341-344.
358, 360
- neutral weak 343
- Noether 383-384
- weak hypercharge 342-344
- weak isospin 342-344
- Yang-Mills 365
Cutoff 219,264-265
CVC hypothesis 320
Cyclotron fonnula 8
d
Dmeson 47
D, meson 47
d quark 39
D'A1embertian 2«1
Darkenergy 414-417
Dark matter 414-417
Davis, R. 27-28. 390.3%
De Broglie wavelength 6
Decay 2.65. 75-84, 87. 197-198
- b quark 351
- c qu;ork 351
- gO) 410
- t:. 77
- 'I,
298. 306
-A
77-78. 324
- Higgs 404-405. 419
- bon 325
- leptoni< 325
- muon 25-29, 75, lIO-315,
394-395,400
)94
25-29. 76,80,315-311,324
- neutrino
- neutron
1 443
"'I Indu
OKay (",nld.)
- nonleptonic
325-326
- n- n-78
- 4> 8)
- pion (charged)
25-29. 7G. 80. 103.
138, 321-324, 349,394-395
- pion (neutral) 65, 138, 143,222
- positronium 256-261
- proton 32-33, 85,406, 4U!, 420
- '"
82-84
- quukonium 298
- semiieplonic 325-326
- tritium 25, 3%
- two-body 24-27. 112. 206-208.
429-430
- Sa also Golden rule: Lifetime
- zII
337. 350-351
Decaymode 79,198
OKay nt'" 197-198, 429-430
Decuplet 36.40, 122-123
Deep inebstic sC<lttering 42-43, SO, 68
Degeneracy 163
Oelbruck scattering 86
to huyon 33, 133
Delu. function
- Dir.ac 20S. 423-427
- Kronecker 155
�nsity of states 203
- See ..Iso Pluse space
DESY 73
�t:liled balance 23, 149
Detector 7-8,203
Deuteriwn 169
Deuteron 43, 103. 130-131, 194.388
Di.;ograrn
- disconnWed 218
- Feynman
60-64, 213
- higher�rder
62-63,75, 86,212.
217-221. 2G2-267
- loop
70. 166-167,217-218, 263,
288, 298-299
- penguin
- trl'!'
148, 157,410
157,217, 265
Difterenti;d cross section
201-203, 223,
430-431
- Su a/w Scattering
Dimensions
- �mplitude 211
- coupling constant 213
- cross 5eCtiOn 211
- field 357
Dipole function 284
Dipole moment
- electric 150,411
- magnetic 150.165, 168, 189-190
Diquark 294, 408
Dinc, P. A. M_ 15.21. 165,219
Dinc delta function 205,423-427
Dirac equ�tion 21. 225-229. 356
- momentum. sp.Ke 234
Dirac Lagr.lgnian 355-356
Dirac matrices 228.434-435
Dirac neutrino 28, 396
Dirac sea 11. 230
Dirac spinor 229
Direct CP viobtion 148
Disconnected diagr.lm 218
Discretesymmetry 118, 136-151.375
Dot product 95
Down quark 39
Downness 49
Drell, S. D. 128,233
Dresden, M. 221
,
Effectivecharge 68-70. 266
Effective mass 135-136, 180, 220
Eigenstate 318
Eigennlue 126-127, 160
Eigenvector Wi-l27
Eight b,aryon problem 133
Eightfold Way 35-37.39,133
Einstein, A. 16,84.417
Einstein summation convention 93
Elastic collision 101-102
Elasticscattering 101-101,199
Electric dipole moment 150, 411
Electric form factor 283-284
Electrodynamics 59,84,238
- Su"Iso Quantum electrodynamics
Electromagnetic current 238,341-344, 358,
360
Eledromagnetic decay 255-261
Electromagnetic field 239
EI�omagnetic force 59
Electromagnetic potential 239
Electron 4-5,13, ISO
Electron gun 4
Electron neutrino 29
Electron number 29.49
Electron volt 9
Electron-electron scattering 246-247. 266.
m
Electron-muon scattering
2-45-246,
252-255, 265, 271
Electron-positron sattering
- elastic 247-248
- inelastic 256-261. 266. 275-279
/"du
Electron-positron �nnihibrion
256-261
altering 255, 279-283
Electron-proton s
Electrowe�k inter�ction
3,43, 60,84,338
- See Qiso GWS theory
Electroweak interference
337
345-346
Electrowe�k unifiGition 338-346
Electroweak mixing
Elementary p,articles: see Particles
Energy
- conservation of 24, 64,80, 87,
100-101. 117, 213, 244
99
- oper.ltor 160,268
- relativistic 96-100
- rest 99
- kinetic
Energy·momentum four·vector
'I meson
98
31, 177-179
'1'meson 41, 177-179
I), meson 298. 306
Euler-u.gr;mge�u.;otion 354-35{j
Event rate 203
i
_ spnor
355-357
356-359
Field strength tensor 238
Fifth force 285
Fine structure 165-166,169-170
Fine structure const�nt to,64, 162. 165, 266
Fine tuning 412
FbvOt 39,43,49, 57,70, 82.129-136,
184-188, 397
Flavor dyn�mics 59-60
Fbvoreigenstates 324.328-329. 397-398
Force 59-60
- dectrom�gneric 59
- gravity 59
- strong 18-20, 30, l4. SO, 59
- we�k 34.48,59
Form factor 282-284, 303,319,322
Four·momentum 98
Four·vector 92-96
- charge"CUrrent 238
- covariant 94
- vector
Exclu.ngeofp,articles 18.61, 183
- contravariant
bclusion principle
- current
1, 3.21. 37. 43-44. 122.
238
- energy·momentum.
130. 183-184
199
botic atom 169
botic p<lrticle 41
Exclusive
-lighilike
Expansion ofuniverse 416-417
- proper velocity
- spacelike 95
ExtelTl<ll line
63,65.213, 243.287,437
413-414
Extra dimensions
f
f. (pion decay constant) 322-323
Faithful representation
119
288
Fermi, E. 24. 60.133. 204, 315
Fermi constant (GF) 313-315
Fermi's Golden Rule 203-211
Fermi theory ofbet<! dec<ly 24. 47-48. 60
Fennibb 6,8,47,395,41)3
Fermion 121-122, 141,183
Feynman. R. P. 21, 29.60. 65. 166.205.219
Feynmanc.akulus 64, 197-223, 373
Feynman diagnm 60-64, 213
Feynman niles 3, 64, 203-204, 369-372.
437-440
- ABC theory 211-214
- GWS theory 307-308. 331
- QCD 283-288
- QED 241-245
- weak interactions 307-308. 331
Feynman-Sttikdberginterpretation 21. 230
Field theory 354-358
- SGilar 355, 357
Fedcev-Popov ghost
94
98
9S
- position·rime 92
%
- timdike 9S
97
Free u.gnngian 369
Free qu�rk 41-44,72
ft·v;r.lue 319
Four·vdocity
Fundamental represent�rion
119, 179
Furry's theorem
273
Fusion 388, 403-404
•
G: su G·parity
GF: sa Fermi constant
332. }46
371
g.. 308, 315, 332, 346
g. 332, 346
Gal;>eticrotation 414-415
Gamma matrices 228
Gamma ray 5,17
g.
g,
yl
- Sa ,,/so Photon
236-237
Gauge field
360. 364-367. 407
117. 304
- abelian 365
- broken 136, 375
- global 358, 363
Gauge invariance
1445
446 1 'nd«
Gauge inl':lriance (comd.)
- local
3. 358-361.363, 367
365
- nonabelian
Gauge transformation
407
134
- SU(,,)
1I8-119, 153
- U(,,)
353-381
Gauge theory
- SU(5)
- SU(6)
239, 270, 360
118-119,153,361, 366
Group theory
117,133
GUT: 5U Grand Unification
Gaussian units
9
Gell·Mmn, M.
34-37,56-57.60,82,133,
Gell·Mmn-Nishijm
i a formula
GUT scale
130-131,
3.48. SO.
60,71-7<),84,221, 332, 337-346
286-287, 367,384
Gell-Mann matrices
405-406, 413
GWS (Glashow/Weinberg/Salam)
145-147
Gyromagnetic ratio
165
156,343
Gel\·Mann-Okubo formula
Generation
Ghost
56-57
h
Hadron
49-51. 75
GIM mechanism
Glashow, S. L
77,327-329
44-48,60, 77,84, 327, 330,
Hadroniution
Half·life
Halo
338,406
Global transformation
Glueball
30
- Su abo Baryon; Me-son
241, 288, 381
358. 363
275-276
79.222,319
415-417
Hamiltonian
160, 268
Hard·sphere scattering
50.67
200-202
Gluino
52
Heniside-Lorentz units
9, 245
Gluon
50,59-60.67, 368
Heaviside step function
205, 426-427
- octet
Heavy lepton
285
67,70, 286, 288
Gluon-gluon coupling
Golden R.u1e
203-211
- for deays
204-20S
- for sc.;ottering
Goldstone boson
208-211
377-381
Goldstone'S theorem
377
47
Heavy neutrino
Heavyquark
Heisenberg. W.
Helidty
3%
159
28, 137-138,234, 241, 268, 324, 339
129
Heisenberg uncertilinty principle
Hermitian conjugate
G·parity
143-144,157
Hermitian matrix
Gradient
22&. 268
Hierarchy problem
Grand Unification
33, 52, 84-85, 405-409
Gr.lvit:ltional lensing
Gr.lvity
415
18, 49, 59.86,413
Gr.lviton
18,49. 59-60,414
Gr«k index
92-93
Greenberg. O. W.
Ground st:lte
43
- finite
- Lie
390,393-394
221, 330, 372. 409
415
15, 159,162-168
Hydrogen
Hypercharge
36, 343-344
Hyperfine splitting
118
- baryons
liS
- mesons
118
- hydrogen
- 0(,,)
lIS-119. 153
- SO(2)
376
- SOP)
llS-1l9, 12S
167-168
180
169-170
33
407
- SO(,,)
11S-119, 153
Identical partides
- SU(2)
128, 130, U5, 362-363. 365
ILC
- SU(3)
128,133, 179,187,367, 407
3«, 346.366. 407
167-168
191
- positronium
Hyperon
- SO(10)
- SU(2)\ x U(1)
62-63, 75. 86. 212,
21
Hot dark matter
118
118
- loren\x.
412,416
Higgsino
't Hooft. G.
lIS
- infinite
84,338, 372. 378-381.
401-402
Home-st:lke mine
118
- discrete
Higgs mechanism
Hole theory
117-118
- abelian
412
6, 50, 52, lSO-3S1, 4{1l-405
217-221, 262-267
373
- continuous
362-363.384
Higher.order proce-ss
- Su also Vacuum
Group
Higgs boson
6.56,80
235, 367
1, 43,66, 183-184. 198
6,401
Impact parameter
Inclusive
199
199-201
1m/ex
- pfOC,J 356-3S7
Jndetenniruocy 3
IndistinguisJu.bte particles
I. •3. 66.
Inel�$tic. sattering
Inertial fram�
199
89
47_<43. 59_60
Intermediat� vector boson
- Sualso W; Z
Internal line 63. 243
Internal momenu 2U-214
142
Intem.t1 symmetry 117. 119. 129-110.411
Intrinsic angular momentum 120
Inlerl;«ting storage rings S
Invarianc� 117
Invariant 93-96. 1()4-107. 110
Inverse beU d«ay 27. 309. 3S8
Inverse Lor�ntz transformation 90. 109
Inversion 139
Inverted spectrum 396
loniution 4.8
Irreducible representation 119
lsospin 129-136. 156. 365
- weak 341-344
Isotope IS
Internal qlUnlUm number
Isotopic spin: su lsospin
j
J/. meson: su • meson. Charmonium
let
SO. 276
lordan form
384
•
K meson 31. 34. 145-ISI
- KI. K.! 146-1.7
- Kt. Ks 147-149
K�on; _ K meson
KEK 43. 47.1<43
Ket
3. 121
Kinetic energy 99. 101. 373
K1�in-Gordon eqlUtiOn
K1ein-Nishina formula
225-227. 240. 35S
272
Kobayashi-MaskaWiJ matrix: su CKM matrix
KM m.atrix: su CKM mattix
KronKktr ddta
Lagrangian
155
357
353-354
3S5-356. 383
_ Fi�ld theory 354-358
- Free 369
- Interaction ].69
_ Klein-Gordon 3SS. 383
- M_ll 3S7-36O
- classical
- Dirac
}66-}68
358
- R�ati¥i$tic 35.-358
_ Supersymmetric 412.420
- Yang-Mills
36S
- Yukawa 38S
ugr.lIlgian density 35•. 357
- QCD
- QED
183-18-4
- Su ..IS<! Lagrangian
H.I66
18.166. 170. 194-195.220.266
1\ (cosmological consunl) 417
A (QCD sale) JOI
A baryon 31-34
A malrilc 93
A, b.uyon .6-47
.I. matrices 286-287
undscape 414
1..elIding log �pproxim..tion 299
Lederman. L M. '19. 1.7, 402
�. T. D. 60, !}6, 142
Left·handed 28. 137-138. 324. 339-342
Left·handed doubl�t 343-344
Leptogmesis 411
Lepton 3, 19,30,45,47.49,122
_ generations 10.•9
- number 28.81, <408-409
- uble JO,49
_ �ak n
i teractions 307. 310, 329-H7
Lamb. W. E.
umbshift
- Sa also Electron: Muon; N�utrino: T�u
Leptonk deay 325
Leploquuk 407-408
Levi·Civita symbol ISS. 253, 271
tHC 6, 47. 52.401. 403. 412-413,416-.17
lifetime 79_80, 91, 197_198,212
_ II
214-215
- channonium 44
- kaon 147
- muon 313-315
_ neutron
319-321
- (1- 57
- pion 323-324
_ positron
ium 171,261. 420
- proton 32-33. <406, <108, 420
- sun 387,399
- ze 51, H7. 350-351
- Su.uso Deay
159. 176
159, 181-193
Li&htquarkm�on IS9, 176-180
Lightlih 9S
Uncar conidcr 5
Unear·plus-Coulomb potenti.al 173
LightqlUrk
Lightqu�rkbaryon
1447
"'I '"dex
Local gau8� n
i v;r;rianc�
- virtu;ol p;lrtid�
loc;ol gauge mmsfonnation
- WIMP
3, 358-363
367
Local phase invariance 360
Logarithmic di�rgence 219,263
Longitudinal polari:wotion 138, 307. 380
Loop diagram 70. 166-167, 263, 288,
298-299
Lorentz gauge 240, 286. 307
Lorentzcontndion 91-92
Lorentz group
liS
Lorentz traos[onnations
89-93
6S
416
Massive gauge field
372, 402
2, 21'-30, 9'9-100,
138-139, 241. 308, 359-361. 377, 380, 392
Matrix elem�n\ 203
Maner-antimaner asymmetry 21,23,51,
148,409-411
Maxim;ol pilrity violation 137, 148, 309
Massless pilrtide
Maxwell, J, c.
84
Maxwell tagrangi,m
l<Jwering operator
IS6
Luminosity 202-203, 261
MaxwelJ's equations
m
Mediator
Mechanics
357-359
238-239, 357-358
2
18,47-48, 59-60, 67, 122, 307,
315,407
Mtheory 414
Magnetic form factor
283-284
Magnetic moment ISO, 165
- anomalous 18, 266
- b.aryons 89-190, 195-1%
- electron 165
- proton 168
Magnetic monopole
Majonna neutrino
409
28,273. 3%. 407
Mandelst;am \t,I;irabl� 113
Marsho..k, R. E. 19
Mass fonnubs
- baryons
191-193,195-1%
57
- Gell·Mann-Okubo 56-57
- Higgs 403
- mesons 179-180.195
- pion 56
- quarkonium 171-176
- WandZ 331
Mns SI
- b.ne 58, 135-136,220
- constituent 58, 135-136, 180, 190,
193
- effective 135-136, 180,220
- eigenstate 328, 391, 31'7-398
- Higgs 403,412
- matrix 362
- neutral kaon 147
- neutrino 31'5-3%
- nonconservation 101-103
- origin 381,402
- physiC<lI 211'-220
- quark 135-136, 180
- relativistic 99, 101
- r�norm;oli� 67,220-221,266
- running 220
- sh�U 65, 213
- tenn
372-375
- Coleman-Glashow
- See "Ise Gluon; Graviton; In\ermediat�
vector boson; Photon; Pioo; W; Z
Meson
15, 18-20, 30-31, 35, 40, 44,
122-123, 176-180
- mass 171'-180
- non�t 31'-'H
- octet 35-36,41
Metric 93-94, 252
Millikan, R. A, 16,41
Minimal coupling 360
Minkowski metric 93-94, 228
Mirrorimag� 136-137,144, 148
Missingmass 415-416
Mixed t�nsor 96
Mixing
fIJ/Bo 145, 149
o
- KO/K 391-394, 399
_
- matriJr:
329,397-398
391-394
- �utral mesons 145, 149
MNS matrix 397-398
Moller scattering 61, 246
Momentum 97
- conservation 64, SO, 97-101, 117,
213, 244
- four-vector 98
- opuitor 226
- relativistic 96-100
- spilc� 234, 370
Mott scanering 245-246, 254-255
M5M 402-406,419
MSW �ffect 392
Multipl�t 130
Multiplicativ� quantum number 141, 143
Multiplicity 130
Muon 19,26-30,310-315
Muon decay 25-29, 75, 310-315, 394-395,
- neutrino
'"
Muon n�utrino
29
'mlu
Muon number
Muonium
29,49
"
N: sa Nucl...,n
21, 230
72-74, 308,
416
IS, 23-30, 1J7-1l9, 389, 415-416
Neutrinoless double beta-deay 396
Neutrino-electron SGltlering 72-73
Neutrino-nucleon SGlrtering n
Neutron 15, 22. 29, 76, 129-130, 150,
315-321
Neutron electric dipole moment
411
285-286, 368
Noether's th...,rem 116-117, 383_384
Nombelian gauge 365
Nonet )9-41
Nonkptonicdeuy
77,325-326
Nonn:diution
- Dine spinor
- Pauli spinor
233. 237, 2&9
126
- polariution vector
- wave funo::tion
November revolution
Nucleon 119-130
24),348
]60
44-47
- Ste..Iso Neutron: Proton
Nucleon-nucleon sattering 131-132
Nucleus 14-15,18,42, 56, 167
0(,,)
118-119,153
35,41, 179
- See ..1so Baryon; Color. Gluon; Meson
a- baryon 37-38, 57,88, 111
'" meson 31, 177-179
Oppenheimer, /' R,
19
Orthogonal matrix 1 ]8-119
Orthogonal pobriutions 242, 348
Orthogonal spinon
Oscill�tion$
OZI rule
268
390-392
82-84, 88, 17<4
P
P:. sa Parity
- meson
141
- oper;!ltor, oper;!ltor
Neutrino m�ss 395-396
Neutrino oscilbtions 30, 52, 74, 387-398
Octet
136-1<42, 157
- conservation 7),1<4]-142, 309
- fermion 141
- invariance 116, 139
]9
329-)38, l43
•
33-3<4, 1<45_1<47
- baryon 1<41
- boson 141
Neutm weak ntfi�ction
i
Ninth gluon
62,2<46
P�rily
Neg.ative enerSl' sl<ite
Neutrino
PaiT production
P�is, A.
", sa Neutron
Neutralino
2J. 62, 65, 170-171. 2<46,
256-261, 29-4-298
]69
u: see Neutrino
NNdermeyer, S.
Ne'em�n, Y. 35
Pair annihilation
1J9-1<41, 236
- particle/antip;orticle
- photon 141
141
- qIQrk 141
- violation 73. 136-1l9, 142, 148, 309
Particle 1<47
-a
5, 14, 156
- antineutrino
24, 27,128
- antineutron 21
- antiproton 21
- B meson 47, 148-149
- B, meson 47
- B, meson 47
- " (bottom) qu;.rk
47
- fJ: sa Electron: Positron
- ' (ch�nn) qIQrk 44
- Dmeson 47
- D, meson 47
- d (down) qu;.rk 39
- IJ. baryon l3. III
- deutfion 4), 103, 130-131. ]94, J.88
- electron 1l, 150
- I) meson 31, 177-179
- ,,'meson
41, ]n-179
- ", �on 298, 306
- y: see Photon
- gtoon SO
- graviton 18,49,59-60,414
-J orJI+: see ", meson
- K meson 31
- �rged
31
- neurral
31, 145-149, 151
- bon: see K meson
- A baryon 31-3<4, n
- A� baryon
- A, baryon
(neutrino)
- /I. (muon)
- u
47
<46-47
19,26-30, 310-315
15,23-30,1l7-H9,
389, 415-416
- neutron 15,76, 129, ISO, 315-321
- nucleon ]29-130
- O- wryon 37-38, 57, n,88, 111
- Of baryon
47
/ '"
450
l ind/!}(
Particle (con/d.)
- wm<':SOn 31, 177-179
- ¢; m<':SOn 31, 83. 177-179
- photon 15-18, 49, 59-60
- 11: meson 19-20. 26-29. 178.
321-324
- pion: su " m<':SOn
- positron 5, IS, 21-22
- proton 4.15. 32-33
- oJ! meson 44. 82-84, 174-175
- S", a/S() Charmonium
- p meson 31, 178
- s (strange) quark 39
- 1: lnryon 33-}4
- 1:� baryon 47
- 1:< baryon 47
- I (top) quark 47
- r l<':pton 47
- .. (up) quark 39
- Y meson 47, SS, 175-176, 195
- S", also Bottomonium
- W 48-49
- 3 baryon 33-}4
- 3) baryon 47
- 3, baryon 47
-Z
48-49, 350-351
10-11, 47
PIlrtick Physics Booklet
Parton 44
- Set illS() Gluon; Quark
Pauli. W. 15, 24, 139.219.227
Pauli (spin) Imltrices 127. ISS. 228. 363.
433-4}4
1. 3,21. 37.
43-44, 122. 130. 183-184
PCAC hypothesis 320,326
Penguin diagram 148,157,410
Pentaquark 43
Perturbation theory 373
Phase transformation 3�-360
Phasespate 203-205. 209. 323-324
'" meson 31. 177-179
Photino 52,412
Photoelectric effect 16, 18
Photon 15-18, 49, 59-60,238-241
Pion 19-20, 26-29, �. 178, 321-324
Pion decay constant 322-323
Planck, M. 15-16
Planck formula 16,100,163
Planck's constant 6, 16
Planck scale 413,420
Plane wave 231. 240
Point particle 220
PoLuvector 139-141
Polariution 138, 147, 307
Pauli (exclusion) principle
Polarization vector 240. 242-243, 286. 307
PontecOlvo, B. 390
Positron 5, IS, 21-22, 230
Positronium 169-172,195-196
Potential
- Coulomb 162,202
- four·vector 239
- linear·plus·CouIomb 173.195
- quark-quark 173, 289-294
- vector 239
Pot/:ntial energ)' 159-160.373
Powell, C. F. 19-20, 25-26
ppchain 3SS-390
Primitive V<':rtI:X 60-61, 64-67, 72-75.
78-80, 212,308
Principal quantum number 163
Proc:a equation 225, 356-357
Projection operator 339, 349
Propagator 213, 369-370.437
- electron 244
- gluon 288
- modified 263. 336
- photon 244
- quark 288
- spin zero 213. 370
- spin one-half 244, 370
- spin one
- massive
308, 370
244, 371
- massless
- unstable particle 336
- Wand Z 308, 333. 336
Proper time 96
Proper velocity %-97
Proton 4.15, 32-33. 129-130
Pseudoscal;o.r 140-141, 236-237
Pseudoscalarm<':SOn 35-36. 122-123.177
Pseudovector 139-141.237
oJ! meson 44, 174-175
- Set Il/'" Charmonium
q
Q: sa Charge, electric
QCD (quantum chromodynamits)
3, 66-71,
173. 283-JOI, 366-369
QED (quantum electrodynamics)
3, 60-66,
165. 225-273
Quantum 16, 19. 369
Quantum fieldtheory 2. 10, 17. 20-21. 52,
122. ISO. 159
2, 17, 20, 100,
120. 159
Quark 3. 37.45.49, 122
- b (bottom) 47
- , (charm) 44-45
Quantum mechanics
Index
- confi�m"'nt 42-44,71-72
- constitu",nt 58
Rochester, G. D. 30-31
Rosenbluth fonnula 283, 303
- d {down) 39
- f� 41-44,72
- m�sses 51, 58, 135-136
- mod",] 37-44,47
- num�r 81
- p�rity 141
- s",� 320
- s"'�n:h 41-44
- 5 (strang"') 39
- I (top) 47
- u (up)
39
- tab]", 49
- val",nc", 320
i luactions 324-337
- w",�k n
Rotation 117-118
Rotation curve 414-415
- Su.me. p�rticl"'$
Quuk·gluon plasm�
71
Qwrk-quark int"'r�ction
Quarkonium
289-294
169, 171-176
- Su aha Bottomonium: Channonium
174-175.195
Quasi·boundmt",
,
Rotation group 118-119.128.154
Rotation m�trix 154_155
Rubbi�. C.
48, 332
Running coupling constant
68.84-85. 220.
265-266, 301. 405-406, 412
Running mass 220
Rutherford, E. 14-15. 42-43
Rutherford sOl.ttering 14-15,42-43. 202.
245-246, 254-255
Ryd�rg ronnula 163
,
S (strangeness)
34-35,40-41
5 (strange) quark 39
Salam. A. 48.60,84,330,338
Sakharov, A. D. 409
Scalar
123. 237
- Su also Invariant
Scmr product
R 278-280
Rabi. l. l. 30. 163
R.oIdi;o] equ�tion 161
R.oIdiativ", corr""tion 166
R�ising operator 156
R.oIng'" 18-19,48, >6, 285
Rank 95-96
R",actor 5, 27
R",al particl", 63.65
R..,(Iuc..,(l mass 169
R",fI""tion 139
R"'8ulariz;otion
219
R",]ativistic cOTr""tion 165
Rdativistic ",n",rgy 98-99
Re!�tivistic mass 99
Relativistic m""hanics 2
Relativistic momentum 98
Rd�tivistic system 159-160
Rdativity 2. 89-113
Reines, F. 27
Renonn�lization
219-221, 262-267.
298-301
Repr�ntation 119,128,199
Resonance 133, 156.279
Rest energy 99. 102
Review ofPartidt Phyrics 10-11
p meson 31. 178
Richter, B. 44
Right·hand..,(l 28. 137-138,324.
339-l4O, 342
94-95
2, 197-203.430-431
- A + A .... B + B 215-217
- Bhabha 62.86, 246
- Compton 17-18, 62,86,113, 246,
249. 271-272
- cross section 132-133
- deep inelastic 42-43
- Iklbrucl< 86
- elastic 199
- eledron-e]""tron 246, 266.271
- electron-muon 245-246, 252_255.
265. 271
- electron-positron 246-248,
256-261, 275-279. 335-337
- electron-proton 255, 279-283
- gIuon-gluon 306
- hard spher'" 200-202
- inelastic 199
- Moller 61,246
- Mott 245-246, 254-255
- neutrino·electron 309-310,
330-334,392-393
- nucleon-nucleon 131-132
- pion-nucleon 132-133.156
- Rutherford 14-15,42-43,202.
245-246, 254-255
SOI.ttering
- S•• also Collisions: Golden Rule
SOI.ttering amplitud'" 132-133
Scattering angle
Scatt",ring ""nter
17, 199-203
200-201
1451
..S2! ,ndex
Schr&linger equation
Step fWlction
159-162,
225-227,230
Schwinger , l , S ,
60,165-166, 219
205, 426-427
Sterile neutrino
395
Stevenson, E. C.
19
Screming
68-69, 85, 167, 266, 299
Storage ring
Se� quark
320
Strange particle
See-s�w mech<mism
SelKtron
StrangencsHhanging
Semileptonic deay
75,325-326, 350
t b�ryon
33-34
Simultaneity
Singl<"l
String theory
52.413-414
411
Strong force
18-20, 30, 34, SO. 129
Strongph�se
410
Stroctureconstant
9
287.304, 368
StUckelberg, E. c. G.
90-91
39,41, 125, 131, 168, 178-179, 187
325
19
Strong CP violation
160-161
293
SI units
30-35
34-35,40-41, 49. 130, 131
Street. I. C.
217
Separation ofvuiablcs
Sextet
Strangeness
396
52
Self.energy
5
Subgroup
21,33
133
SLAC
4, 42,44, 47, 51. 148, 174
Subquark: set Substructure
Slash
249, 252-253
Substructure
Slepton
52,411
SO(II)
Supermultiplet
118-119, 153
Solar neutrino
50,52,58,417
SuperKamiokande
Supernova
394
394, 396, 400
Sol�r neutrino problem
387-390
Superstring
Solar neutrino spectrum
389
Supersymmetry
Sneutrino
SNO
393-394
Sp�celike
SU(,,)
Symmetry
412
Synchrotron radiation
73
Spin 1/2
T: setTime reversal
, (top) quark
125-128
Spin down
125,231,234
Spin matrix
126-128,234,339
Spin and statistics
168,191
229
125, 128
Tensor
3, 149-151
402
95-96, 237
8c= see ubibbo angle
Tevatron
6,47.401. 403
f4: set weak mixing angle
Spontaneous symmetry.breaking
375-378,
Theory of Everything
Thomson,l· J.
Three·jet ev<:nt
Threshold
6,52
Standard Modd
32, 79-80
3, "9-52, 135-136,
392
Standard Solar Model
390
204,209
311. 316
276
23,87, 103, 106-107. 11 L 278
Time dependence
160,230.390
Time dilation
91-92, %
Time reversal
149-151
Timelike
Ting, C. C.
3, 122
S\.;Itistical factor
Sl, 4!3
13_14
Three-body decay
52, 412
S\.;Ible particle
State
Technicolor
125,128
- Dirac
49
141-142
TCPtheorem
165, 168
SSC
r-8 puzzle
250
Spin-spin coupling
- Pauli
47
47
\.;Iu number
125,231, 234
Spin-orbit coupling
""
r lepton
122, 183
Spin-averaged amplitude
Squark
5
161
120-121
Spinor
1l0-1i1
115-lSI
15, 162-164, 172
Spherical harmonic
Spin up
122.125. 183-184
Symmetric tensor
7
Spec\.;llor quark
Spin
118
Symmetric state
9S
Spectrum
6,52. 85, 402.406, 411-41l,
SUSY: st" Supersymmetry
201-203
Spark chamber
Sparticle
52,85. 413-414
416,420
52,412
Solid angle
392-395
37. 133
Top
47
95
44
Index
Toponium 47
Total cross section
Tr�c�
VKtor
251-252
237
- Su also Four �tor
202
Vector interaction 341
Trac� theor�ms 252-253, 271, 435
Vector meson
Transformation
Vector pot�ntial 239
Velocity
- ordinary %-97,105
- proper 96-97
V�locity addition 91-92
Vertex 60, 212, 369, 371. 438-440
- charg� coniug�tion 269
- Dir�c spinor 235, 269
- four·�tor 94
- Lor�ntz 89-93
- plIrity 236
- tensor 95
Transition probability
Tr�nsl�tion 117-118
Transpose 118.235
Trms�rs� pol�rization 241
Treediagnm 157.217.265
Triangl� function 112
Triangle group 117-118. 153
Triplet 125. 131, 168. 178
Tritium
25, 56. 3%
Truth 47,49
- Su a/sa Top
21<m lin� 168
Two-bodydK�y 24-27. 112, 206-208.
429-430
Two·body sanering 209-211. 430-431
Two-i"'t��nt 276
Two-n�utrino hypothesis 29
"
" (up) quark 39
U(n)
118
Ultraviolet utastroplu: 15
Unc�rta.inty principl� 6. 56,80
Unification 11-4-85. 413
- Su also Electroweak; GUT; GWS
Unitarity triangl� 352.410
Unitary matrix 118-119, 350,352,
362-363. 367
Units 9-10.211. 357
Upn�ss 49
T meson 47,88. 175-176. 195
- Su also Bonomonium
,
V-A int�raction 309, 320. 331
V�nts 31
Vacuum 167.373-375
Vacuum �xpectation va.lu� 403. 417
Vacuum polariUltion 69-70, 85,167,
262-267, 299
Valence quark 320
Van der Waals forc� 70
- ABC
122-123,177
213
- QCD 66,288-289, 371-372
- QED 60-61. 244.371
- weak 308.325, 329, 331-3)2
Vertex corredion 217
Vertex factor: 5te Feynmm rules; Vertex
Virial theorem 159
Virtual particle 63,65.213
•
W boson 48-49, 59-60. 88, 307-308
Ward identity 267
Wav� function 3. 160. 162. 181-188.
195,226
Weal< contamination 334
Weak coupling constant 308. 315
W�akcunent 341-342
Weak eigenstate 397
- Su allo CKM matrix, MNS matrix
Weak force 34.48.71-79
W�ak hypercharge 342-344
W�ak interaction
71-79. 136-137. 307-346
- ch�rged: su Charged weak interaction
- neutr3l: su Neutral weak interaction
Weak i$OSpin 342-344
Weak mixing angl� 51. 332
Weak phase 410
Weinberg. S. 48,60,84,330, 338
Weinberg angle: suWeak mixing angle
Weyt. H. 139.361. 365
WIMP 416
Wino
52.412
Work function 16
Wu, C. S. 136-137
,
X mediator
S b.oryon
407-408
33-34
y
Y medi�tor 407-408
Yang, c. N. 60,136, 142, 361
Yang-Mills theory 361-366, 384
1453
·s.
l inda
Yukawa, H.
19, -48-49, 71
Yukawa coupling
Yukawa meson
•
Zboson
38I, 18S.402
15,19,56
-48-.9. 51, 59-60, 72.
307-308
Z decay
337, 350-351
Z factory
335
Z pole
335-337
Zino
52, 412,416
Zumino. B.
411
Zweig, G.
37,83